## Subtraction

Subtraction is ....

... taking one number away from another.

## Try It Yourself

Train yourself.

You get good at subtraction with practice...

... so use Math Trainer - Subtraction to train yourself!

## Subtraction Table

You can also "look up" answers for simple subtraction using this table:

## Example: Find 8 − 5

- find the row starting with "8"
- move along till you are under the column "5"
- and there is the number "3", so 8 − 5 = 3

Other names used in subtraction are Minus, Less, Difference, Decrease, Take Away, Deduct.

The names of the numbers in a subtraction fact are:

Minuend − Subtrahend = Difference

Minuend : The number that is to be subtracted from.

Subtrahend : The number that is to be subtracted.

Difference : The result of subtracting one number from another.

## Subtracting Larger Numbers

To subtract numbers with more than one digit (such as "42−25") use any of these methods, choose the one you prefer :

For practice try these Subtraction Worksheets

## Subtraction

Subtraction is the process of taking away a number from another. It is a primary arithmetic operation that is denoted by a subtraction symbol (-) and is the method of calculating the difference between two numbers.

## What Is Subtraction?

Subtraction is an operation used to find the difference between numbers . When you have a group of objects and you take away a few objects from it, the group becomes smaller. For example, you bought 9 cupcakes for your birthday party and your friends ate 7 cupcakes. Now you are left with 2 cupcakes. This can be written in the form of a subtraction expression: 9 - 7 = 2 and is read as "nine minus seven equals two". When we subtract 7 from 9, (9 - 7) we get 2. Here, we performed the subtraction operation on two numbers 9 and 7 to get the difference of 2.

## Subtraction Symbol

In mathematics, we have different symbols. The subtraction symbol is one of the important math symbols that we use while performing subtraction. In the above section, we read about subtracting two numbers 9 and 7. If we observe this subtraction: (9 - 7 = 2), the symbol (-) connects the two numbers and completes the given expression. This symbol is also known as the minus sign.

## Subtraction Formula

When we subtract two numbers, we use some terms which are used in the subtraction expression:

- Minuend: The number from which the other number is subtracted.
- Subtrahend: The number which is to be subtracted from the minuend.
- Difference: The final result after subtracting the subtrahend from the minuend.

The subtraction formula is written as: Minuend - Subtrahend = Difference

Let us understand the subtraction formula or the mathematical equation of subtraction with an example.

Here, 9 is the minuend, 7 is the subtrahend, and 2 is the difference.

## How To Solve Subtraction Problems?

While solving subtraction problems, one-digit numbers can be subtracted in a simple way, but for larger numbers, we split the numbers into columns using their respective place values , like Ones, Tens, Hundreds, Thousands, and so on. While solving such problems we may encounter some cases with borrowing and some without borrowing. Subtraction with borrowing is also known as subtraction with regrouping. When the minuend is smaller than the subtrahend, we use the regrouping method. While regrouping, we borrow 1 number from the preceding column to make the minuend bigger than the subtrahend. Let us understand this with the help of a few examples.

## Subtraction Without Regrouping

Example: Subtract 25632 from 48756.

Note: In subtraction, we always subtract the smaller number from the larger number to get the correct answer.

Solution: Follow the given steps and try to relate them with the following figure.

Step 1: Start with the digit at ones place. (6 - 2 = 4) Step 2: Move to the tens place. (5 - 3 = 2) Step 3: Now subtract the digits at hundreds place. (7 - 6 = 1) Step 4: Now subtract the digits at thousands place. (8 - 5 = 3) Step 5: Finally, subtract the digits at ten thousands place. (4 - 2 = 2) Step 6: Therefore, the difference between the two given numbers is: 48756 - 25632 = 23124.

## Subtraction With Regrouping

Example: Subtract 3678 from 8162.

Solution: Follow the given steps and try to relate them with the following figure. We need to solve: 8162 - 3678 Step 1: Start subtracting the digits at ones place. We can see that 8 is greater than 2. So, we will borrow 1 from the tens column which will make it 12. Now, 12 - 8 = 4 ones. Step 2: After giving 1 to the ones column in the previous step, 6 becomes 5. Now, let us subtract the digits at the tens place (5 - 7). Here, 7 is greater than 5, so we will borrow 1 from the hundreds column. This will make it 15. So,15 - 7 = 8 tens. Step 3: In step 2 we had given 1 to the tens column, so we are left with 0 at the hundreds place. To subtract the digits on the hundreds place, i.e., (0 - 6) we will borrow 1 from the thousands column. This will make it 10. So, 10 - 6 = 4 hundreds. Step 4: Now, let us subtract the digits at the thousands place. After giving 1 to the hundreds column, we have 7. So, 7 - 3 = 4 Step 5: Therefore, the difference between the two given numbers is: 8162 - 3678 = 4484

## Subtraction Using Number Line

A number line is a visual aid that helps us understand subtraction because it allows us to jump backward and forward on each number. To understand how this works, let us explore subtraction using a number line. Let us subtract 4 from 9 using a number line. We will start by marking the number 9 on the number line. When we subtract using a number line, we count by moving one number at a time towards the left-hand side. Since we are subtracting 4 from 9, we will move 4 times to the left. The number on which you land after 4 backward jumps, is the answer. Thus, 9 - 4 = 5.

## Real Life Subtraction Word Problems

The concept of subtraction is often used in our day-to-day activities. Let us understand how to solve real-life subtraction word problems with the help of an interesting example.

Example: A soccer match had a total of 4535 spectators. After the first innings, 2332 spectators left the stadium. Find the number of remaining spectators.

Solution: Given: The total number of spectators present in the first innings = 4535; The number of spectators who left the stadium after the first innings = 2332 Here, 4535 is the minuend and 2332 is the subtrahend.

Th H T O 4 5 3 5 -2 3 3 2 2 2 0 3

Therefore, the number of remaining spectators = 2203.

Important Notes on Subtraction:

Here are a few important notes that you can follow while performing subtraction in your everyday life.

- Any subtraction problem can be transformed into an addition problem and vice-versa.
- Subtracting 0 from any number gives the number itself as the difference.
- When 1 is subtracted from any number, the difference equals the predecessor of the number.
- Words like "Minus", "Less", "Difference", "Decrease", "Take Away" and "Deduct" indicate that you need to subtract one number from another.

## Topics Related to Subtraction

Check out these interesting articles to know about subtraction and its related topics.

- Binary Subtraction
- Subtraction Calculator
- Addition and Subtraction of Fractions
- Subtraction of Complex Numbers
- Subtraction of Fractions

## Subtraction Examples

Example 1: In an International cricket match, Sri Lanka scored 236 runs and India scored 126 runs. How many more runs should India score to be equal to the number of runs scored by Sri Lanka?

Runs scored by Sri Lanka = 236; Runs scored by India = 126 To find the number of runs that India should score more to be equal to the number of runs scored by Sri Lanka, we will subtract 126 from 236.

H T O 2 3 6 - 1 2 6 1 1 0

Therefore, India must score 110 more runs to be equal to Sri Lanka's runs.

Example 2: Jerry collected 189 seashells and Eva collected 54 shells. Who collected more seashells and by how much?

Number of shells collected by Jerry = 189; Number of shells collected by Eva = 54

This shows that Jerry collected more seashells. Let us subtract 189 - 54 to get the difference.

H T O 1 8 9 - 0 5 4 1 3 5

Therefore, Jerry collected 135 seashells more than Eva.

Example 3: During an annual Easter egg hunt, the participants found 2469 eggs in the clubhouse, out of which 54 Easter eggs were broken. Can you find out the number of unbroken eggs?

The number of easter eggs found in the Clubhouse = 2469; Number of easter eggs that were broken = 54; The total number of unbroken eggs=?

Now, we will subtract the number of broken eggs from the total number of eggs.

Th H T O 2 4 6 9 - 5 4 2 4 1 5

Therefore, the number of unbroken eggs are 2415.

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## Practice Questions on Subtraction

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## FAQs on Subtraction

Where do we use subtraction.

Subtraction is used in our day-to-day life. For example, if we want to know how much money we spent on the items that we bought, or, how much money is left with us, or, if we want to calculate the time left in finishing a task, we use subtraction.

## What Are the Types of Subtraction?

The types of subtraction mean the various methods used in subtraction. For example, subtraction with and without regrouping, subtraction using number charts, subtraction using number line, the subtraction of small numbers using you fingers, and so on.

## What Are Subtraction Strategies?

Subtraction strategies are different ways in which subtraction can be learned. For example, using a number line, with the help of a Place Value Chart, separating the Tens and Ones and then subtracting them separately, and many others.

## Give Some Subtraction Examples.

There can be various real-life examples of subtraction. For example, if you have 5 apples and your friend ate 3 apples. Using subtraction, we can find out the number of remaining apples: 5 - 3 = 2. So, 2 apples are left with you. Similarly, if there are 16 students in a class, out of which 9 are girls, then we can find out the number of boys in the class by subtracting 9 from 16. (16 - 9 = 7). So, we know that there are 7 boys in the class.

## What Are the Three Parts of Subtraction?

The 3 parts of subtraction are named as follows:

- Minuend: The number from which we subtract the other number is known as the minuend.
- Subtrahend: The number which is subtracted from the minuend is known as the subtrahend.
- Difference: The final result obtained after performing subtraction is known as the difference.

## How Do You Write a Subtraction?

While writing subtraction, the two important symbols are '-' (minus) and '=' (equal to). The minus sign means when one number is being subtracted from the other number. And the equal to sign delivers the final result.

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## Solving Equations Using Subtraction

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## Solving One-Step Linear Equations: Adding & Subtracting

Add/Subtract Times/Divide Multi-Step Parentheses Zero/No/All Sol'n

"Linear" equations are equations with just a plain old variable like " x ", rather than something more complicated like x 2 , or x / y , or square roots, or other more-complicated expressions. Linear equations are the simplest equations that you'll deal with.

You've probably already solved linear equations; you just didn't know it. Back in your early years, when you were learning addition, your teacher probably gave you worksheets to complete that had exercises like the following:

Fill in the box: □ + 3 = 5

Once you'd learned your addition facts well enough, you knew that you had to put a " 2 " inside the box.

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Solving equations works in much the same way, but now we have to figure out what goes into the x , instead of what goes into the box. However, since we're older now than when we were filling in boxes, the equations can also be much more complicated, and therefore the methods we'll use to solve the equations will be a bit more advanced.

In general, to solve an equation for a given variable, we need to "undo" whatever has been done to the variable. We do this in order to get the variable by itself; in technical terms, we are "isolating" the variable. This results in the equation being rearranged to say "(variable) equals (some number)", where (some number) is the answer they're looking for. For instance:

## Solve x + 6 = –3

The variable is the letter x . To solve this equation, I need to get the x by itself; that is, I need to get x on one side of the "equals" sign, and some number on the other side.

Since I want just x on the one side, this means that I don't like the "plus six" that's currently on the same side as the x . Since the 6 is added to the x , I need to subtract this 6 to get rid of it. That is, I will need to subtract a 6 from the x in order to "undo" their having added a 6 to it.

This brings up the most important consideration with equations:

No matter what kind of equation we're dealing with — linear or otherwise — whatever we do to the one side of the equation, we must do the exact same thing to the other side of the equation. Equations are like toddlers in this respect:

We have to be totally, totally fair to the two sides, or unhappiness will ensue!

Whatever you do to an equation, do the EXACT SAME thing to BOTH sides of that equation!

Probably the best way to keep track of this subtraction of the 6 from both sides is to format your work this way:

The above image is animated on the "live" page.

What you see here is that I've subtracted 6 from both sides, drawn a horizontal "equals" bar underneath the entire equation, and then added down. On the left-hand side (LHS) of the equation, this gives me:

x plus nothing is x , and 6 minus 6 is zero

On the right-hand side (RHS) of the equation, I have:

–3 plus – 6 is – 9

The solution is the last line of my work; namely:

x = –9

The same "undo" procedure works for equations in which the variable has been paired with a subtraction.

## Solve x – 3 = –5

The variable is on the left-hand side (LHS) of the equation, and it's paired with a "subtract three". Since I want to get x by itself, I don't like the " 3 " that's currently subtracted from it. The opposite of subtraction is addition, so I'll undo the "subtract 3 " by adding 3 to both sides of the equation, and then adding down to simplify to get my answer:

Then my answer is:

x = –2

You may be instructed to "check your solutions", at least in the early stages of learning how to solve equations. To do this "checking", you need only plug your answer into the original equation, and make sure that you end up with a true statement. (This is, after all, the definition of the solution to an equation; namely, the solution is any value, or set of values [for more complicated equations, later on], which makes the original equation a true statement.)

So, to check my solution to the above equation, you'd plug " –2 " in place of the x in left-hand side (LHS) of the original equation, and check that this simplifies to give the original value for the right-hand side (RHS) of the equation:

LHS: (–2) – 3 = –5

RHS: –5

Because each side of the original equation now evaluates to the exact same thing, this confirms that the solution is indeed correct.

## Solve 4 = x – 3 , and check your solution.

This time, the variable is on the right-hand side (RHS) of the equation. That's okay; it doesn't matter where the variable is, as long as I get isolate it (that is, as long as I can get it by itself on one side of the "equals" sign).

In this equation, I've got a three that's subtracted from the variable. To undo the subtraction, I'll add three to either side of the equation.

4 = x - 3 +3 + 3 ---------- 7 = x

(I could have written the right-hand side, after adding down, as " x + 0 ", but "plus zero" is customarily ignored. That's why I carried down only the x on the right-hand side.)

Now, as part of my hand-in work, I need to show that I've checked this solution by plugging it back into the RHS of the original equation, and confirming that I end up with the LHS of the original equation; that is, that I end up with 4 :

RHS: (7) – 3 = 4 = LHS

The "checking" part is what I just did above. I've made sure to label things clearly, so the grader is able to find my "check" (so I'll get full credit on the exercise). My final answer is:

When I solved the last exercise above, the variable had ended up on the right-hand side of the "equals" sign. But in my solution, I wrote the answer with the variable on the left-hand side of the "equals" sign. This is pretty standard. When you're solving, the variable will end up wherever it ends up. When you're writing out the solution, the variable goes on the left. Why? Because.

## Solve 2 = – x

This equation is almost solved. But not quite. I don't have plain old x on the right-hand side; instead, I've got – x . What to do?

I can kind-of think of the – x as being 0 – x . So what would happen if I added x to each side of the equation?

2 = –x +x +x ------- x + 2 = 0

Okay; that helped. By taking the variable and "adding it over to the other side", I've now got the variable in a format I like. And this has also converted the original equation into a simple one-step equation. I'll get rid of the 2 from the left-hand side by "subtracting it over to" the right-hand side:

This answer makes sense. If the negative of the variable equalled a positive two, then the positive of the variable should equal a negative two. So my answer is:

Technically, that last example was a two-step equation, because solving it required adding one thing to both sides of the equation, and then subtracting another thing to both sides. The important thing to notice is that you can add and subtract variables to the other side of an equation, just like you can add and subtract numbers to the other side. The exact same methods work with both variables and numbers.

You can use the Mathway widget below to practice solving a linear equation by adding or subtracting. Try the entered exercise, or type in your own exercise. Then click the button to compare your answer to Mathway's. (Or skip the widget and continue with the lesson.)

(Click "Tap to view steps" to be taken directly to the Mathway site for a paid upgrade.)

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## Solving Algebra Equations with Addition and Subtraction

- Always perform the same operation to both sides of the equation.
- You can add and subtract numbers from both sides of the equation to solve for x or y.
- Always double check you answer by plugging it back into the original equation.

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Subtraction is a foundational math skill that students need to become comfortable with performing. Math Games enables children from pre-K through 7th grade to have loads of fun as they practice this skill in accessible, adventure-filled games!

With our site’s free, curriculum-based games, apps, digital textbook, and downloadable worksheets, pupils are able to easily review tasks such as:

- Subtracting with pictures and whole numbers
- Completing subtraction sentences
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What’s more, Math Games lets kids enjoy themselves as they learn! Our popular resources are available to use in the classroom or at home, and can be easily tailored to suit different grade levels and abilities. Click on a skill above to start using our tools!

## Subtraction Table – Definition, Chart, Facts, Examples, FAQs

What is a subtraction table in math, how to create a subtraction table, patterns in the subtraction table, solved examples on subtraction table, practice problems on subtraction table, frequently asked questions about subtraction table.

A subtraction table in math is a tabular representation of numbers arranged in rows and columns that enables us to determine the difference between two numbers only via observation rather than through calculation.

Subtraction is one of the four basic arithmetic operations in mathematics. Subtraction in math is a process of finding the difference between two numbers. Let’s quickly understand important terms in a subtraction equation.

- Minuend : The first number, the number from which another number will be subtracted
- Subtrahend : The second number, the number to be subtracted

Identifying the subtrahend and the minuend will help in reading and writing the subtraction table.

The subtraction table makes the process of subtraction easier. It is formed by arranging the numbers in rows and columns in a certain pattern.

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## Subtraction Table (Subtraction Chart)

Using the subtraction table, we can calculate the difference between two numbers without doing the calculation.

A basic subtraction chart takes you through the subtraction of numbers from 1 to 10.

Here, look for the first number in the top-most row and the second number in the first column.

Let’s try a different approach. We can also write the numbers from 0 to 12. Also, if we change the positions where the minuend and subtrahend are written in the table, we will look for the first number in the first column, and the second number in the top-row.

Take a look at the subtraction table build this way:

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## How to Read a Subtraction Table Chart

Let’s take an example to understand how to read a subtraction table:

Ex: 6 – 9

Step 1: Look first for a number in the horizontal row. (Here, we need to find number 6 in the horizontal row)

Step 2: subtract from that number by selecting a number in the leftmost column (Here, we need to find number 9 in the FIRST column)

Step 3: Move along the row and down the column to find the cell where they meet. The number written in the cell is the answer. The answer here is -3.

Let us write a subtraction table for 1 to 12.

Step 1: T he first cell (top-left corner) shows the operation. So, we write the minus sign (-) here.

Step 2: The top-most row represents the minuend or the first number. This is where we look for the first number when we read the chart. Here, we write the numbers from 12 to 1. (You can also write the numbers from 1 to 12.)

Step 3: The first column (the left-most column) represents the subtrahend or the second number. This is where you search for the second number when you read the chart. Write the numbers from 1 to 12.

Step 4: Now, we fill in the cells based on the row and column it represents. We identify the minuend (the topmost number) and the subtrahend (the leftmost number) for the given cell. Subtract and write the difference in the given cell.

When filling the first row, the subtrahend (second number) is 1 (it stays the same for the entire row). We keep changing the minuend (the first number) using the top-most row and fill each cell accordingly.

Calculations for the first row:

12 – 1 = 11

10 – 1 = 9

9 – 1 = 8

8 – 1 = 7

7 – 1 = 6

6 – 1 = 5

5 – 1 = 4

4 – 1 = 3

3 -1 = 2

2 – 1 = 1

1 – 1 = 0

Thus, the entries of the first row are 11, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 0.

In this way, we can create a subtraction table on our own. Here’s a subtraction table for numbers 1 to 10. You can try creating a subtraction table 1-100 for practice.

## Subtraction Facts in a Subtraction Table

Let’s write the individual subtraction facts of 1-12 using the subtraction table.

Observe the subtraction table closely and you will find many interesting patterns formed due to the properties of subtraction.

We know that subtracting the same number from itself results in 0. So, the diagonal elements from top-right to the bottom-left are 0.

Also, subtracting a greater number from a smaller number results in a negative number. Thus, the half triangular section below the diagonal has negative integers. The triangular section above the diagonal has positive integers.

## Facts about Subtraction Table

- Subtraction tables help with understanding the basics of subtraction, patterns in subtraction.
- Zero subtracted from any number gives the number itself.
- 1 subtracted from a number gives the predecessor of the number as the difference. For example: 150 – 1= 149
- The number from which the other number is to be subtracted is called the minuend. The number subtracted is called the subtrahend and the answer is called the difference.
- Minuend – Subtrahend = Difference

In this article, we have learnt about the subtraction table and how to read and write it. Now let’s solve some examples and practice problems to understand the concept better

Example 1: Complete the following subtraction table by finding the missing numbers.

12 – 2 = 10

10 – 2 = 8

11 – 3 = 8

Example 2: How will you find the value of 10 – 2 by reading the subtraction table? Explain with steps.

Solution:

Step 1: Look for the first number in the horizontal row.

Step 2: Find the second number in the leftmost column.

Step 3: Move along the row and down the column to find where they meet. The number you arrive at the intersection is the answer. The answer here is 8.

Example 3: Peter has 12 marbles. He gave 5 away to friends. How many marbles are left? Use a subtraction chart to solve the problem.

Here, we have to find 12 – 5.

Let’s find it by using a subtraction chart.

Look first for number 12 in the horizontal row.

Now subtract from that number by selecting number 5 in the leftmost column.

The row and column meet at the cell where we find the answer.

The answer here is 7.

Thus, Peter has 7 seashells left.

## Subtraction Table - Definition, Chart, Facts, Examples, FAQs

Attend this quiz & Test your knowledge.

## In the equation below, identify which number is the subtrahend. 50 - 8 = 42

Subtracting ____ from any number results in the same number., the number from which we subtract another number is known as _____., which of the following patterns is visible in the subtraction table.

Does the order of numbers matter in subtraction of real numbers?

In the case of subtraction, the order of factors is important because the subtraction is not commutative for real numbers.

What are the properties of subtraction ?

1. Not commutative: Changing the order of numbers affects the result.

2. Subtracting two natural numbers does not always result in a natural number (for example: 5- 8 = -3)

3. Subtracting 0 from any number gives the same number as a result because it is a neutral element.

What are subtraction table guidelines?

Understand the positions of the minuend and the subtrahend based on how the table is created (top-row and first column). The answer can be found where the column and row meet.

What are the ways to teach Subtraction?

Following are the ways to teach subtraction:

a) Use real life objects that you can find around the house.

b) Repeat everything twice, thrice and more ! This may seem a bit boring at first, but it is really necessary.

c) Use a ton of visual examples so that children can remember everything more easily.

In subtraction, what happens when the subtrahend and minuend are equal to each other?

When the subtrahend and minuend are equal, the resulting answer is zero.

In subtraction, is it possible that the subtrahend is larger than the minuend?

Yes, either one of the minuends or the subtrahend can be larger than the other in an equation. The placement of both numbers will ultimately impact the difference.

What are subtraction facts?

Subtraction facts are simple numerical expressions stating the difference between two numbers. For example, 9 – 2 = 7 and 8 – 2 = 6 are examples of subtraction facts of 2.

## RELATED POSTS

- Minus – Definition with Examples
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## 2.1: Solve Equations Using the Subtraction and Addition Properties of Equality

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## Learning Objectives

By the end of this section, you will be able to:

- Verify a solution of an equation
- Solve equations using the Subtraction and Addition Properties of Equality
- Solve equations that require simplification
- Translate to an equation and solve
- Translate and solve applications

Before you get started, take this readiness quiz.

- Evaluate \(x+4\) when \(x=−3\). If you missed this problem, review Exercise 1.5.25 .
- Evaluate \(15−y\) when \(y=−5\). If you missed this problem, review Exercise 1.5.31 .
- Simplify \(4(4n+1)−15n\). If you missed this problem, review Exercise 1.10.49 .
- Translate into algebra “5 is less than x.” If you missed this problem, review Exercise 1.3.43 .

## Verify a Solution of an Equation

Solving an equation is like discovering the answer to a puzzle. The purpose in solving an equation is to find the value or values of the variable that make each side of the equation the same – so that we end up with a true statement. Any value of the variable that makes the equation true is called a solution to the equation. It is the answer to the puzzle!

## Definition: Solution of an Equation

A solution of an equation is a value of a variable that makes a true statement when substituted into the equation.

## TO DETERMINE WHETHER A NUMBER IS A Solution TO AN EQUATION

- Substitute the number in for the variable in the equation.
- Simplify the expressions on both sides of the equation.
- If it is true, the number is a solution.
- If it is not true, the number is not a solution.

## Example \(\PageIndex{1}\)

Determine whether \(x = \frac{3}{2}\) is a solution of \(4x−2=2x+1\).

Since a solution to an equation is a value of the variable that makes the equation true, begin by substituting the value of the solution for the variable.

Since \(x = \frac{3}{2}\) results in a true equation (4 is in fact equal to 4), \(\frac{3}{2}\) is a solution to the equation \(4x−2=2x+1\).

## Try It \(\PageIndex{2}\)

Is \(y = \frac{4}{3}\) a solution of \(9y+2=6y+3\)?

## Try It \(\PageIndex{3}\)

Is \(y = \frac{7}{5}\) a solution of \(5y+3=10y-4\)?

## Solve Equations Using the Subtraction and Addition Properties of Equality

We are going to use a model to clarify the process of solving an equation. An envelope represents the variable – since its contents are unknown – and each counter represents one. We will set out one envelope and some counters on our workspace, as shown in Figure \(\PageIndex{1}\). Both sides of the workspace have the same number of counters, but some counters are “hidden” in the envelope. Can you tell how many counters are in the envelope?

What are you thinking? What steps are you taking in your mind to figure out how many counters are in the envelope?

Perhaps you are thinking: “I need to remove the 3 counters at the bottom left to get the envelope by itself. The 3 counters on the left can be matched with 3 on the right and so I can take them away from both sides. That leaves five on the right—so there must be 5 counters in the envelope.” See Figure \(\PageIndex{2}\) for an illustration of this process.

What algebraic equation would match this situation? In Figure \(\PageIndex{3}\) each side of the workspace represents an expression and the center line takes the place of the equal sign. We will call the contents of the envelope x.

Let’s write algebraically the steps we took to discover how many counters were in the envelope:

Five in the envelope plus three more does equal eight!

Our model has given us an idea of what we need to do to solve one kind of equation. The goal is to isolate the variable by itself on one side of the equation. To solve equations such as these mathematically, we use the Subtraction Property of Equality .

## SUBTRACTION PROPERTY OF EQUALITY

For any numbers a , b , and c ,

\[\begin{array} {ll} {\text{If}} &{a = b} \\ {\text{then}} &{a - c = b - c} \end{array}\]

When you subtract the same quantity from both sides of an equation, you still have equality.

Doing the Manipulative Mathematics activity “Subtraction Property of Equality” will help you develop a better understanding of how to solve equations by using the Subtraction Property of Equality .

Let’s see how to use this property to solve an equation. Remember, the goal is to isolate the variable on one side of the equation. And we check our solutions by substituting the value into the equation to make sure we have a true statement.

## Example \(\PageIndex{4}\)

Solve: \(y+37=−13\).

To get y by itself, we will undo the addition of 37 by using the Subtraction Property of Equality.

Since y=−50 makes y+37=−13 a true statement, we have the solution to this equation.

## Try It \(\PageIndex{5}\)

Solve: \(x+19=−27\).

\(x=−46\)

## Try It \(\PageIndex{6}\)

Solve: \(x+16=−34\).

\(x=−50\)

What happens when an equation has a number subtracted from the variable, as in the equation \(x−5=8\)? We use another property of equations to solve equations where a number is subtracted from the variable. We want to isolate the variable, so to ‘undo’ the subtraction we will add the number to both sides. We use the Addition Property of Equality .

## ADDITION PROPERTY OF EQUALITY

\[\begin{array} {ll} {\text{If}} &{a = b} \\ {\text{then}} &{a + c = b + c} \end{array}\]

When you add the same quantity from both sides of an equation, you still have equality.

In Exercise \(\PageIndex{4}\), 37 was added to the y and so we subtracted 37 to ‘undo’ the addition. In Exercise \(\PageIndex{7}\), we will need to ‘undo’ subtraction by using the Addition Property of Equality .

## Example \(\PageIndex{7}\)

Solve: \(a−28=−37\).

## Try It \(\PageIndex{8}\)

Solve: \(n−61=−75\).

\(n=−14\)

## Try It \(\PageIndex{9}\)

Solve: \(p−41=−73\).

\(p=−32\)

## Example \(\PageIndex{10}\)

Solve: \(x - \frac{5}{8} = \frac{3}{4}\)

## Try It \(\PageIndex{11}\)

Solve: \(p−\frac{2}{3}=\frac{5}{6}\).

\(p = \frac{9}{6} p =\frac{3}{2}\)

## Try It \(\PageIndex{12}\)

Solve: \(q−\frac{1}{2}=\frac{5}{6}\).

\(q =\frac{4}{3}\)

The next example will be an equation with decimals.

## Example \(\PageIndex{13}\)

Solve: \(n−0.63=−4.2\).

## Try It \(\PageIndex{14}\)

Solve: \(b−0.47=−2.1\).

\(b=−1.63\)

## Try It \(\PageIndex{15}\)

Solve: \(c−0.93=−4.6\).

\(c=−3.67\)

## Solve Equations That Require Simplification

In the previous examples, we were able to isolate the variable with just one operation. Most of the equations we encounter in algebra will take more steps to solve. Usually, we will need to simplify one or both sides of an equation before using the Subtraction or Addition Properties of Equality.

You should always simplify as much as possible before you try to isolate the variable. Remember that to simplify an expression means to do all the operations in the expression. Simplify one side of the equation at a time. Note that simplification is different from the process used to solve an equation in which we apply an operation to both sides.

## Example \(\PageIndex{16}\): How to Solve Equations That Require Simplification

Solve: \(9x−5−8x−6=7\).

## Try It \(\PageIndex{17}\)

Solve: \(8y−4−7y−7=4\).

## Try It \(\PageIndex{18}\)

Solve: \(6z+5−5z−4=3\).

## Example \(\PageIndex{19}\)

Solve: 5(n−4)−4n=−8.

We simplify both sides of the equation as much as possible before we try to isolate the variable.

## Try It \(\PageIndex{20}\)

Solve: \(5(p−3)−4p=−10\).

## Try It \(\PageIndex{21}\)

Solve: \(4(q+2)−3q=−8\).

\(q=−16\)

## Example \(\PageIndex{22}\)

Solve: \(3(2y−1)−5y=2(y+1)−2(y+3)\).

We simplify both sides of the equation before we isolate the variable.

## Try It \(\PageIndex{23}\)

Solve: \(4(2h−3)−7h=6(h−2)−6(h−1)\).

## Try It \(\PageIndex{24}\)

Solve: \(2(5x+2)−9x=3(x−2)−3(x−4)\).

## Translate to an Equation and Solve

To solve applications algebraically, we will begin by translating from English sentences into equations. Our first step is to look for the word (or words) that would translate to the equals sign . Here are some of the words that are commonly used.

- is equal to
- is the same as
- the result is

The steps we use to translate a sentence into an equation are listed below.

## TRANSLATE AN ENGLISH SENTENCE TO AN ALGEBRAIC EQUATION

- Locate the “equals” word(s). Translate to an equals sign (=).
- Translate the words to the left of the “equals” word(s) into an algebraic expression.
- Translate the words to the right of the “equals” word(s) into an algebraic expression.

## Example \(\PageIndex{25}\)

Translate and solve: Eleven more than x is equal to 54.

## Try It \(\PageIndex{26}\)

Translate and solve: Ten more than x is equal to 41.

\(x+10=41;x=31\)

## Try It \(\PageIndex{27}\)

Translate and solve: Twelve less than x is equal to 51.

y−12=51;y=63

## Example \(\PageIndex{28}\)

Translate and solve: The difference of 12t and 11t is −14.

## Try It \(\PageIndex{29}\)

Translate and solve: The difference of 4x and 3x is 14.

\(4x−3x=14;x=14\)

## Try It \(\PageIndex{30}\)

Translate and solve: The difference of 7a and 6a is −8.

\(7a−6a=−8;a=−8\)

## Translate and Solve Applications

Most of the time a question that requires an algebraic solution comes out of a real life question. To begin with that question is asked in English (or the language of the person asking) and not in math symbols. Because of this, it is an important skill to be able to translate an everyday situation into algebraic language.

We will start by restating the problem in just one sentence, assign a variable, and then translate the sentence into an equation to solve. When assigning a variable, choose a letter that reminds you of what you are looking for. For example, you might use q for the number of quarters if you were solving a problem about coins.

## Example \(\PageIndex{31}\): How to Solve Translate and Solve Applications

The MacIntyre family recycled newspapers for two months. The two months of newspapers weighed a total of 57 pounds. The second month, the newspapers weighed 28 pounds. How much did the newspapers weigh the first month?

## Try It \(\PageIndex{32}\)

Translate into an algebraic equation and solve:

The Pappas family has two cats, Zeus and Athena. Together, they weigh 23 pounds. Zeus weighs 16 pounds. How much does Athena weigh?

## Try It \(\PageIndex{33}\)

Sam and Henry are roommates. Together, they have 68 books. Sam has 26 books. How many books does Henry have?

## SOLVE AN APPLICATION.

- Read the problem. Make sure all the words and ideas are understood.
- Identify what we are looking for.
- Name what we are looking for. Choose a variable to represent that quantity.
- Translate into an equation. It may be helpful to restate the problem in one sentence with the important information.
- Solve the equation using good algebra techniques.
- Check the answer in the problem and make sure it makes sense.
- Answer the question with a complete sentence.

## Example \(\PageIndex{34}\)

Randell paid $28,675 for his new car. This was $875 less than the sticker price. What was the sticker price of the car?

\(\begin{array} {ll} {\textbf {Step 1. Read}\text{ the problem. }} &{}\\\\ {\textbf {Step 2. Identify}\text{ what we are looking for.}} &{\text{"What was the sticker price of the car?"}} \\\\ {\textbf{Step 3. Name}\text{ what we are looking for.}} &{} \\ {\text{Choose a variable to represent that quantity.}} &{\text{Let s = the sticker price of the car.}} \\\\{\textbf {Step 4. Translate}\text{ into an equation. Restate }} &{} \\ {\text{the problem in one sentence.}} &{$\text{28675 is } $\text{875 less than the sticker price}} \\ \\ {} &{$\text{28675 is } $\text{875 less than s}}\\ {}&{28675 = s - 875} \\ {\textbf {Step 5. Solve}\text{ the equation. }} &{28675 + 875 = s - 875 + 875}\\ {} &{29550 = s} \\ \\ {\textbf {Step 6. Check}\text{ the answer. }} &{} \\ {\text{Is }$875\text{ less than }$29550\text{ equal to } $28675?} &{} \\ {29550 - 875 \stackrel{?}{=} 28675} &{} \\ {28675 = 28675\checkmark} &{} \\ \\ {\textbf {Step 7. Answer}\text{ the question with }} &{\text{The sticker price of the car was }$29550.} \\ {\text{a complete sentence.}} &{} \end{array}\)

## Try It \(\PageIndex{35}\)

Eddie paid $19875 for his new car. This was $1025 less than the sticker price. What was the sticker price of the car?

## Try It \(\PageIndex{36}\)

The admission price for the movies during the day is $7.75. This is $3.25 less the price at night. How much does the movie cost at night?

## Key Concepts

- For any numbers a , b , and c , if a=b, then a+c=b+c.
- For any numbers a , b , and c , if a=b, then a−c=b−c.
- Locate the “equals” word(s). Translate to an equal sign (=).

## Solver Title

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- Pre Algebra Order of Operations Factors & Primes Fractions Long Arithmetic Decimals Exponents & Radicals Ratios & Proportions Percent Modulo Number Line Mean, Median & Mode
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- Pre Calculus Equations Inequalities Scientific Calculator Scientific Notation Arithmetics Complex Numbers Polar/Cartesian Simultaneous Equations System of Inequalities Polynomials Rationales Functions Arithmetic & Comp. Coordinate Geometry Plane Geometry Solid Geometry Conic Sections Trigonometry
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- x^4-5x^2+4=0
- \sqrt{x-1}-x=-7
- \left|3x+1\right|=4
- \log _2(x+1)=\log _3(27)
- 3^x=9^{x+5}
- What is the completing square method?
- Completing the square method is a technique for find the solutions of a quadratic equation of the form ax^2 + bx + c = 0. This method involves completing the square of the quadratic expression to the form (x + d)^2 = e, where d and e are constants.
- What is the golden rule for solving equations?
- The golden rule for solving equations is to keep both sides of the equation balanced so that they are always equal.
- How do you simplify equations?
- To simplify equations, combine like terms, remove parethesis, use the order of operations.
- How do you solve linear equations?
- To solve a linear equation, get the variable on one side of the equation by using inverse operations.

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## Algebra Worksheets

Welcome to the Algebra worksheets page at Math-Drills.com, where unknowns are common and variables are the norm. On this page, you will find Algebra worksheets for middle school students on topics such as algebraic expressions, equations and graphing functions.

This page starts off with some missing numbers worksheets for younger students. We then get right into algebra by helping students recognize and understand the basic language related to algebra. The rest of the page covers some of the main topics you'll encounter in algebra units. Remember that by teaching students algebra, you are helping to create the future financial whizzes, engineers, and scientists that will solve all of our world's problems.

Algebra is much more interesting when things are more real. Solving linear equations is much more fun with a two pan balance, some mystery bags and a bunch of jelly beans. Algebra tiles are used by many teachers to help students understand a variety of algebra topics. And there is nothing like a set of co-ordinate axes to solve systems of linear equations.

## Most Popular Algebra Worksheets this Week

## Algebraic Properties, Rules and Laws Worksheets

The commutative law or commutative property states that you can change the order of the numbers in an arithmetic problem and still get the same results. In the context of arithmetic, it only works with addition or multiplication operations , but not mixed addition and multiplication. For example, 3 + 5 = 5 + 3 and 9 × 5 = 5 × 9. A fun activity that you can use in the classroom is to brainstorm non-numerical things from everyday life that are commutative and non-commutative. Putting on socks, for example, is commutative because you can put on the right sock then the left sock or you can put on the left sock then the right sock and you will end up with the same result. Putting on underwear and pants, however, is non-commutative.

- The Commutative Law Worksheets The Commutative Law of Addition (Numbers Only) The Commutative Law of Addition (Some Variables) The Commutative Law of Multiplication (Numbers Only) The Commutative Law of Multiplication (Some Variables)

The associative law or associative property allows you to change the grouping of the operations in an arithmetic problem with two or more steps without changing the result. The order of the numbers stays the same in the associative law. As with the commutative law, it applies to addition-only or multiplication-only problems. It is best thought of in the context of order of operations as it requires that parentheses must be dealt with first. An example of the associative law is: (9 + 5) + 6 = 9 + (5 + 6). In this case, it doesn't matter if you add 9 + 5 first or 5 + 6 first, you will end up with the same result. Students might think of some examples from their experience such as putting items on a tray at lunch. They could put the milk and vegetables on their tray first then the sandwich or they could start with the vegetables and sandwich then put on the milk. If their tray looks the same both times, they will have modeled the associative law. Reading a book could be argued as either associative or nonassociative as one could potentially read the final chapters first and still understand the book as well as someone who read the book the normal way.

- The Associative Law Worksheets The Associative Law of Addition (Whole Numbers Only) The Associative Law of Multiplication (Whole Numbers Only)

Inverse relationships worksheets cover a pre-algebra skill meant to help students understand the relationship between multiplication and division and the relationship between addition and subtraction.

- Inverse Mathematical Relationships with One Blank Addition and Subtraction Easy Addition and Subtraction Harder All Multiplication and Division Facts 1 to 18 in color (no blanks) Multiplication and Division Range 1 to 9 Multiplication and Division Range 5 to 12 Multiplication and Division All Inverse Relationships Range 2 to 9 Multiplication and Division All Inverse Relationships Range 5 to 12 Multiplication and Division All Inverse Relationships Range 10 to 25
- Inverse Mathematical Relationships with Two Blanks Addition and Subtraction (Sums 1-18) Addition and Subtraction Inverse Relationships with 1 Addition and Subtraction Inverse Relationships with 2 Addition and Subtraction Inverse Relationships with 3 Addition and Subtraction Inverse Relationships with 4 Addition and Subtraction Inverse Relationships with 5 Addition and Subtraction Inverse Relationships with 6 Addition and Subtraction Inverse Relationships with 7 Addition and Subtraction Inverse Relationships with 8 Addition and Subtraction Inverse Relationships with 9 Addition and Subtraction Inverse Relationships with 10 Addition and Subtraction Inverse Relationships with 11 Addition and Subtraction Inverse Relationships with 12 Addition and Subtraction Inverse Relationships with 13 Addition and Subtraction Inverse Relationships with 14 Addition and Subtraction Inverse Relationships with 15 Addition and Subtraction Inverse Relationships with 16 Addition and Subtraction Inverse Relationships with 17 Addition and Subtraction Inverse Relationships with 18

The distributive property is an important skill to have in algebra. In simple terms, it means that you can split one of the factors in multiplication into addends, multiply each addend separately, add the results, and you will end up with the same answer. It is also useful in mental math, an example of which should help illustrate the definition. Consider the question, 35 × 12. Splitting the 12 into 10 + 2 gives us an opportunity to complete the question mentally using the distributive property. First multiply 35 × 10 to get 350. Second, multiply 35 × 2 to get 70. Lastly, add 350 + 70 to get 420. In algebra, the distributive property becomes useful in cases where one cannot easily add the other factor before multiplying. For example, in the expression, 3(x + 5), x + 5 cannot be added without knowing the value of x. Instead, the distributive property can be used to multiply 3 × x and 3 × 5 to get 3x + 15.

- Distributive Property Worksheets Distributive Property (Answers do not include exponents) Distributive Property (Some answers include exponents) Distributive Property (All answers include exponents)

Students should be able to substitute known values in for an unknown(s) in an expression and evaluate the expression's value.

- Evaluating Expressions with Known Values Evaluating Expressions with One Variable, One Step and No Exponents Evaluating Expressions with One Variable and One Step Evaluating Expressions with One Variable and Two Steps Evaluating Expressions with Up to Two Variables and Two Steps Evaluating Expressions with Up to Two Variables and Three Steps Evaluating Expressions with Up to Three Variables and Four Steps Evaluating Expressions with Up to Three Variables and Five Steps

As the title says, these worksheets include only basic exponent rules questions. Each question only has two exponents to deal with; complicated mixed up terms and things that a more advanced student might work out are left alone. For example, 4 2 is (2 2 ) 2 = 2 4 , but these worksheets just leave it as 4 2 , so students can focus on learning how to multiply and divide exponents more or less in isolation.

- Exponent Rules for Multiplying, Dividing and Powers Mixed Exponent Rules (All Positive) Mixed Exponent Rules (With Negatives) Multiplying Exponents (All Positive) Multiplying Exponents (With Negatives) Multiplying the Same Exponent with Different Bases (All Positive) Multiplying the Same Exponent with Different Bases (With Negatives) Dividing Exponents with a Greater Exponent in Dividend (All Positive) Dividing Exponents with a Greater Exponent in Dividend (With Negatives) Dividing Exponents with a Greater Exponent in Divisor (All Positive) Dividing Exponents with a Greater Exponent in Divisor (With Negatives) Powers of Exponents (All Positive) Powers of Exponents (With Negatives)

Knowing the language of algebra can help to extract meaning from word problems and to situations outside of school. In these worksheets, students are challenged to convert phrases into algebraic expressions.

- Translating Algebraic Phrases into Expressions Translating Algebraic Phrases into Expressions (Simple Version) Translating Algebraic Phrases into Expressions (Complex Version)

Combining like terms is something that happens a lot in algebra. Students can be introduced to the topic and practice a bit with these worksheets. The bar is raised with the adding and subtracting versions that introduce parentheses into the expressions. For students who have a good grasp of fractions, simplifying simple algebraic fractions worksheets present a bit of a challenge over the other worksheets in this section.

- Simplifying Expressions by Combining Like Terms Simplifying Linear Expressions with 3 terms Simplifying Linear Expressions with 4 terms Simplifying Linear Expressions with 5 terms Simplifying Linear Expressions with 6 to 10 terms
- Simplifying Expressions by Combining Like Terms with Some Arithmetic Adding and simplifying linear expressions Adding and simplifying linear expressions with multipliers Adding and simplifying linear expressions with some multipliers . Subtracting and simplifying linear expressions Subtracting and simplifying linear expressions with multipliers Subtracting and simplifying linear expressions with some multipliers Mixed adding and subtracting and simplifying linear expressions Mixed adding and subtracting and simplifying linear expressions with multipliers Mixed adding and subtracting and simplifying linear expressions with some multipliers Simplify simple algebraic fractions (easier) Simplify simple algebraic fractions (harder)
- Rewriting Linear Equations Rewrite Linear Equations in Standard Form Convert Linear Equations from Standard to Slope-Intercept Form Convert Linear Equations from Slope-Intercept to Standard Form Convert Linear Equations Between Standard and Slope-Intercept Form
- Rewriting Formulas Rewriting Formulas (addition and subtraction; about one step) Rewriting Formulas (addition and subtraction; about two steps) Rewriting Formulas ( multiplication and division ; about one step)

## Linear Expressions and Equations

In these worksheets, the unknown is limited to the question side of the equation which could be on the left or the right of equal sign.

- Missing Numbers in Equations with Blanks as Unknowns Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Blanks in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 20 ; Blanks in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 20 ; Blanks in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 20 ; Blanks in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Blanks Never in Answer Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Blanks in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 20 ; Blanks in Any Position )
- Missing Numbers in Equations with Symbols as Unknowns Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Symbols in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 20 ; Symbols in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 20 ; Symbols in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 20 ; Symbols in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Symbols Never in Answer Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Symbols in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 20 ; Symbols in Any Position )
- Solving Equations with Addition and Symbols as Unknowns Equalities with Addition (0 to 9) Symbol Unknowns Equalities with Addition (1 to 12) Symbol Unknowns Equalities with Addition (1 to 15) Symbol Unknowns Equalities with Addition (1 to 25) Symbol Unknowns Equalities with Addition (1 to 99) Symbol Unknowns
- Missing Numbers in Equations with Variables as Unknowns Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Variables Never in Answer Position ) Missing Numbers in Equations ( All Operations ; Range 1 to 20 ; Variables in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( Addition Only ; Range 1 to 20 ; Variables in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( Subtraction Only ; Range 1 to 20 ; Variables in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( Multiplication Only ; Range 1 to 20 ; Variables in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Variables Never in Answer Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 9 ; Variables in Any Position ) Missing Numbers in Equations ( Division Only ; Range 1 to 20 ; Variables in Any Position )
- Solving Simple Linear Equations Solving Simple Linear Equations with Values from -9 to 9 (Unknown on Left Side) Solving Simple Linear Equations with Values from -99 to 99 (Unknown on Left Side) Solving Simple Linear Equations with Values from -9 to 9 (Unknown on Right or Left Side) Solving Simple Linear Equations with Values from -99 to 99 (Unknown on Right or Left Side)
- Determining Linear Equations from Slopes, y-intercepts and Points Determine a Linear Equation from the Slope and y-intercept Determine a Linear Equation from the Slope and a Point Determine a Linear Equation from Two Points Determine a Linear Equation from Two Points by Graphing

Graphing linear equations and reading existing graphs give students a visual representation that is very useful in understanding the concepts of slope and y-intercept.

- Graphing Linear Equations Graph Slope-Intercept Equations
- Determinging Linear Equations from Graphs Determine the Equation from a Graph Determine the Slope from a Graph Determine the y-intercept from a Graph Determine the x-intercept from a Graph Determine the slope and y-intercept from a Graph Determine the slope and intercepts from a Graph Determine the slope, intercepts and equation from a Graph

Solving linear equations with jelly beans is a fun activity to try with students first learning algebraic concepts. Ideally, you will want some opaque bags with no mass, but since that isn't quite possible (the no mass part), there is a bit of a condition here that will actually help students understand equations better. Any bags that you use have to be balanced on the other side of the equation with empty ones.

Probably the best way to illustrate this is through an example. Let's use 3 x + 2 = 14. You may recognize the x as the unknown which is actually the number of jelly beans we put in each opaque bag. The 3 in the 3 x means that we need three bags. It's best to fill the bags with the required number of jelly beans out of view of the students, so they actually have to solve the equation.

On one side of the two-pan balance, place the three bags with x jelly beans in each one and two loose jelly beans to represent the + 2 part of the equation. On the other side of the balance, place 14 jelly beans and three empty bags which you will note are required to "balance" the equation properly. Now comes the fun part... if students remove the two loose jelly beans from one side of the equation, things become unbalanced, so they need to remove two jelly beans from the other side of the balance to keep things even. Eating the jelly beans is optional. The goal is to isolate the bags on one side of the balance without any loose jelly beans while still balancing the equation.

The last step is to divide the loose jelly beans on one side of the equation into the same number of groups as there are bags. This will probably give you a good indication of how many jelly beans there are in each bag. If not, eat some and try again. Now, we realize this won't work for every linear equation as it is hard to have negative jelly beans, but it is another teaching strategy that you can use for algebra.

Despite all appearances, equations of the type a/ x are not linear. Instead, they belong to a different kind of equations. They are good for combining them with linear equations, since they introduce the concept of valid and invalid answers for an equation (what will be later called the domain of a function). In this case, the invalid answers for equations in the form a/ x , are those that make the denominator become 0.

- Solving Linear Equations Combining Like Terms and Solving Simple Linear Equations Solving a x = c Linear Equations Solving a x = c Linear Equations including negatives Solving x /a = c Linear Equations Solving x /a = c Linear Equations including negatives Solving a/ x = c Linear Equations Solving a/ x = c Linear Equations including negatives Solving a x + b = c Linear Equations Solving a x + b = c Linear Equations including negatives Solving a x - b = c Linear Equations Solving a x - b = c Linear Equations including negatives Solving a x ± b = c Linear Equations Solving a x ± b = c Linear Equations including negatives Solving x /a ± b = c Linear Equations Solving x /a ± b = c Linear Equations including negatives Solving a/ x ± b = c Linear Equations Solving a/ x ± b = c Linear Equations including negatives Solving various a/ x ± b = c and x /a ± b = c Linear Equations Solving various a/ x ± b = c and x /a ± b = c Linear Equations including negatives Solving linear equations of all types Solving linear equations of all types including negatives

## Linear Systems

- Solving Systems of Linear Equations Easy Linear Systems with Two Variables Easy Linear Systems with Two Variables including negative values Linear Systems with Two Variables Linear Systems with Two Variables including negative values Easy Linear Systems with Three Variables; Easy Easy Linear Systems with Three Variables including negative values Linear Systems with Three Variables Linear Systems with Three Variables including negative values
- Solving Systems of Linear Equations by Graphing Solve Linear Systems by Graphing (Solutions in first quadrant only) Solve Standard Linear Systems by Graphing Solve Slope-Intercept Linear Systems by Graphing Solve Various Linear Systems by Graphing Identify the Dependent Linear System by Graphing Identify the Inconsistent Linear System by Graphing

## Quadratic Expressions and Equations

- Simplifying (Combining Like Terms) Quadratic Expressions Simplifying quadratic expressions with 5 terms Simplifying quadratic expressions with 6 terms Simplifying quadratic expressions with 7 terms Simplifying quadratic expressions with 8 terms Simplifying quadratic expressions with 9 terms Simplifying quadratic expressions with 10 terms Simplifying quadratic expressions with 5 to 10 terms
- Adding/Subtracting and Simplifying Quadratic Expressions Adding and simplifying quadratic expressions. Adding and simplifying quadratic expressions with multipliers. Adding and simplifying quadratic expressions with some multipliers. Subtracting and simplifying quadratic expressions. Subtracting and simplifying quadratic expressions with multipliers. Subtracting and simplifying quadratic expressions with some multipliers. Mixed adding and subtracting and simplifying quadratic expressions. Mixed adding and subtracting and simplifying quadratic expressions with multipliers. Mixed adding and subtracting and simplifying quadratic expressions with some multipliers.
- Multiplying Factors to Get Quadratic Expressions Multiplying Factors of Quadratics with Coefficients of 1 Multiplying Factors of Quadratics with Coefficients of 1 or -1 Multiplying Factors of Quadratics with Coefficients of 1, or 2 Multiplying Factors of Quadratics with Coefficients of 1, -1, 2 or -2 Multiplying Factors of Quadratics with Coefficients up to 9 Multiplying Factors of Quadratics with Coefficients between -9 and 9

The factoring quadratic expressions worksheets in this section provide many practice questions for students to hone their factoring strategies. If you would rather worksheets with quadratic equations, please see the next section. These worksheets come in a variety of levels with the easier ones are at the beginning. The 'a' coefficients referred to below are the coefficients of the x 2 term as in the general quadratic expression: ax 2 + bx + c. There are also worksheets in this section for calculating sum and product and for determining the operands for sum and product pairs.

- Factoring Quadratic Expressions Factoring Quadratic Expressions with Positive 'a' coefficients of 1 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients of 1 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients of 1 with a Common Factor Step Factoring Quadratic Expressions with Positive 'a' coefficients up to 4 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 4 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 4 with a Common Factor Step Factoring Quadratic Expressions with Positive 'a' coefficients up to 5 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 5 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 5 with a Common Factor Step Factoring Quadratic Expressions with Positive 'a' coefficients up to 9 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 9 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 9 with a Common Factor Step Factoring Quadratic Expressions with Positive 'a' coefficients up to 81 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 81 Factoring Quadratic Expressions with Positive or Negative 'a' coefficients up to 81 with a Common Factor Step Calculating Sum and Product (Operand Range 0 to 9 ) ✎ Calculating Sum and Product (Operand Range 1 to 9 ) ✎ Calculating Sum and Product (Operand Range 0 to 9 Including Negatives ) ✎ Calculating Sum and Product (Operand Range 1 to 9 Including Negatives ) ✎ Calculating Sum and Product (Operand Range -20 to 20 ) ✎ Calculating Sum and Product (Operand Range -99 to 99 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 0 to 9 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 1 to 9 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 0 to 12 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 1 to 12 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 0 to 9 Including Negatives ) ✎ Determining Operands from Sum and Product Pairs (Operand Range 1 to 9 Including Negatives ) ✎ Determining Operands from Sum and Product Pairs (Operand Range -20 to 20 ) ✎ Determining Operands from Sum and Product Pairs (Operand Range -99 to 99 ) ✎

Whether you use trial and error, completing the square or the general quadratic formula, these worksheets include a plethora of practice questions with answers. In the first section, the worksheets include questions where the quadratic expressions equal 0. This makes the process similar to factoring quadratic expressions, with the additional step of finding the values for x when the expression is equal to 0. In the second section, the expressions are generally equal to something other than x, so there is an additional step at the beginning to make the quadratic expression equal zero.

- Solving Quadratic Equations that Equal Zero Solving Quadratic Equations with Positive 'a' coefficients of 1 Solving Quadratic Equations with Positive or Negative 'a' coefficients of 1 Solving Quadratic Equations with Positive or Negative 'a' coefficients of 1 with a Common Factor Step Solving Quadratic Equations with Positive 'a' coefficients up to 4 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 4 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 4 with a Common Factor Step Solving Quadratic Equations with Positive 'a' coefficients up to 5 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 5 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 5 with a Common Factor Step Solving Quadratic Equations with Positive 'a' coefficients up to 9 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 9 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 9 with a Common Factor Step Solving Quadratic Equations with Positive 'a' coefficients up to 81 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 81 Solving Quadratic Equations with Positive or Negative 'a' coefficients up to 81 with a Common Factor Step
- Solving Quadratic Equations that Equal an Integer Solving Quadratic Equations for x ("a" coefficients of 1) Solving Quadratic Equations for x ("a" coefficients of 1 or -1) Solving Quadratic Equations for x ("a" coefficients up to 4) Solving Quadratic Equations for x ("a" coefficients between -4 and 4) Solving Quadratic Equations for x ("a" coefficients up to 81) Solving Quadratic Equations for x ("a" coefficients between -81 and 81)

## Other Polynomial and Monomial Expressions & Equations

- Simplifying Polynomials That Involve Addition And Subtraction Addition and Subtraction; 1 variable; 3 terms Addition and Subtraction; 1 variable; 4 terms Addition and Subtraction; 2 variables; 4 terms Addition and Subtraction; 2 variables; 5 terms Addition and Subtraction; 2 variables; 6 terms
- Simplifying Polynomials That Involve Multiplication And Division Multiplication and Division; 1 variable; 3 terms Multiplication and Division; 1 variable; 4 terms Multiplication and Division; 2 variables; 4 terms Multiplication and Division; 2 variables; 5 terms
- Simplifying Polynomials That Involve Addition, Subtraction, Multiplication And Division All Operations; 1 variable; 3 terms All Operations; 1 variable; 4 terms All Operations; 2 variables; 4 terms All Operations; 2 variables; 5 terms All Operations (Challenge)
- Factoring Expressions That Do Not Include A Squared Variable Factoring Non-Quadratic Expressions with No Squares, Simple Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with No Squares, Simple Coefficients, and Negative and Positive Multipliers Factoring Non-Quadratic Expressions with No Squares, Compound Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with No Squares, Compound Coefficients, and Negative and Positive Multipliers
- Factoring Expressions That Always Include A Squared Variable Factoring Non-Quadratic Expressions with All Squares, Simple Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with All Squares, Simple Coefficients, and Negative and Positive Multipliers Factoring Non-Quadratic Expressions with All Squares, Compound Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with All Squares, Compound Coefficients, and Negative and Positive Multipliers
- Factoring Expressions That Sometimes Include Squared Variables Factoring Non-Quadratic Expressions with Some Squares, Simple Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with Some Squares, Simple Coefficients, and Negative and Positive Multipliers Factoring Non-Quadratic Expressions with Some Squares, Compound Coefficients, and Positive Multipliers Factoring Non-Quadratic Expressions with Some Squares, Compound Coefficients, and Negative and Positive Multipliers
- Multiplying Polynomials With Two Factors Multiplying a monomial by a binomial Multiplying two binomials Multiplying a monomial by a trinomial Multiplying a binomial by a trinomial Multiplying two trinomials Multiplying two random mon/polynomials
- Multiplying Polynomials With Three Factors Multiplying a monomial by two binomials Multiplying three binomials Multiplying two binomials by a trinomial Multiplying a binomial by two trinomials Multiplying three trinomials Multiplying three random mon/polynomials

## Inequalities

- Writing The Inequality That Matches The Graph Writing Inequalities for Graphs
- Graphing Inequalities On Number Lines Graphing Inequalities (Basic)
- Solving Linear Inequalities Solving Inequalities Including a Third Term Solving Inequalities Including a Third Term and Multiplication Solving Inequalities Including a Third Term, Multiplication and Division

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Lesson 1: Basic addition and subtraction Basic addition Add within 5 Basic subtraction Subtract within 5 Add and subtract: pieces of fruit Add within 10 Subtract within 10 Relating addition and subtraction Relate addition and subtraction Math > Arithmetic (all content) > Addition and subtraction > Basic addition and subtraction

Example: Find 8 − 5 find the row starting with "8" move along till you are under the column "5" and there is the number "3", so 8 − 5 = 3 Names Other names used in subtraction are Minus, Less, Difference, Decrease, Take Away, Deduct. The names of the numbers in a subtraction fact are: Minuend − Subtrahend = Difference

k + 22 = 29 We want to get k by itself on the left hand side of the equation. So, what can we do to undo adding 22? We can subtract 22 because the inverse operation of addition is subtraction! Here's how subtracting 22 from each side looks: k + 22 = 29 k + 22 − 22 = 29 − 22 Subtract 22 from each side. k = 7 Simplify. Let's check our work.

In mathematics, we have different symbols. The subtraction symbol is one of the important math symbols that we use while performing subtraction. In the above section, we read about subtracting two numbers 9 and 7. If we observe this subtraction: (9 - 7 = 2), the symbol (-) connects the two numbers and completes the given expression.

x + a − a = x. x + a − a = x. The inverse of subtraction is addition. If we start with a number x and subtract a number a, then adding a to the result will return us to the original number x. In symbols, x − a + a = x. x − a + a = x. Example 5. Solve x − 8 = 10 x − 8 = 10 for x.

Students learn to solve one-step subtraction equations. For example, to solve z - 3 = 16, add 3 to both sides of the equation, to get z = 19. Next, check the solution by substituting a 19 back into the original equation, to get (19) - 3 = 16, which is a true statement, so the solution checks. We help you determine the exact lessons you need.

Solve Equations Using the Subtraction and Addition Properties of Equality. We began our work solving equations in previous chapters. It has been a while since we have seen an equation, so we will review some of the key concepts before we go any further.. We said that solving an equation is like discovering the answer to a puzzle.

First draw a picture of x+7=11. The left side of the equation is x+7. Let's draw an x and a 7 on the left side of our picture. The right side is 11, so let's draw an 11 on the right. Challenge Can you think of a number to replace "x" so that the left side and the right side add up to the same total? Answer

For a complete lesson on subtraction equations, go to https://www.MathHelp.com - 1000+ online math lessons featuring a personal math teacher inside every les...

Solve Equations Using the Addition Property of Equality. In all the equations we have solved so far, a number was added to the variable on one side of the equation. We used subtraction to "undo" the addition in order to isolate the variable. But suppose we have an equation with a number subtracted from the variable, such as x − 5 = 8. x ...

Simply, 3 − 1 = 2 Let's understand the concept with the help of the following example of apples. In the example above, if Harry has 6 apples and he gives 3 apples to Jim, How many apples are left with him? We can calculate this by subtracting 3 from 6: 6 − 3 = 3 Harry is left with 3 apples. Related Worksheets View View View View View View View View

This answer makes sense. If the negative of the variable equalled a positive two, then the positive of the variable should equal a negative two. So my answer is: Technically, that last example was a two-step equation, because solving it required adding one thing to both sides of the equation, and then subtracting another thing to both sides.

7 questions Practice Subtract within 10 7 questions Practice Relate addition and subtraction 7 questions Practice Making 10 Learn Getting to 10 by filling boxes Adding to 10

We can sometimes solve an equation by adding or subtracting the same number to both sides of the equation. We know this is okay, because as long as we perform the same operation to both sides of the equation, then the equation doesn't change. Let's try solving for this simple example by adding or subtracting to both sides: x + 5 = 7

For example, 5 - 3 = 10 - 8 is a subtraction equation in which both sides equal the same thing. 5 - 3 = 2 and 10 - 8 = 2. It's important to know how to work with these subtraction equations ...

Kindergarten K.59 / Subtraction with Pictures Up to 10 K.60 / Subtract Two Numbers Up to 9 K.61 / Choose Subtraction Pictures Up to 10 K.63 / Subtraction Up to 9 K.68 / Subtraction with Pictures Up to 5 K.69 / Subtract Two Numbers Up to 5 K.70 / Choose Subtraction Pictures with Numbers Up to 5 1

Subtraction in math is a process of finding the difference between two numbers. Let's quickly understand important terms in a subtraction equation. Minuend: The first number, the number from which another number will be subtracted; Subtrahend: The second number, the number to be subtracted

Solve Equations Using the Subtraction and Addition Properties of Equality. We are going to use a model to clarify the process of solving an equation. An envelope represents the variable - since its contents are unknown - and each counter represents one. ... To begin with that question is asked in English (or the language of the person ...

If the calculations involve a combination of addition, subtraction, multiplication and division then. Step 1: First, perform the multiplication and division from left to right. Step 2: Then, perform addition and subtraction from left to right. Example: Calculate 9 × 2 - 10 ÷ 5 + 1 =. Solution:

This leaves us with the equation shown below. This is an example of an equation with a negative integer. x = 5 - 4. Step 2: Solve for the subtraction question "5 - 4", this will lead to the answer which is "x = 1". x = 1. Example 2: Integers Subtraction. 4 - x = 5. Step 1: On your paper, write down the equation that is shown above.

Free equations calculator - solve linear, quadratic, polynomial, radical, exponential and logarithmic equations with all the steps. Type in any equation to get the solution, steps and graph

Welcome to the Algebra worksheets page at Math-Drills.com, where unknowns are common and variables are the norm. On this page, you will find Algebra worksheets for middle school students on topics such as algebraic expressions, equations and graphing functions. This page starts off with some missing numbers worksheets for younger students.