## Slope Calculator

## If the 2 Points are Known

If 1 point and the slope are known.

Slope, sometimes referred to as gradient in mathematics, is a number that measures the steepness and direction of a line, or a section of a line connecting two points, and is usually denoted by m . Generally, a line's steepness is measured by the absolute value of its slope, m . The larger the value is, the steeper the line. Given m , it is possible to determine the direction of the line that m describes based on its sign and value:

- A line is increasing, and goes upwards from left to right when m > 0
- A line is decreasing, and goes downwards from left to right when m < 0
- A line has a constant slope, and is horizontal when m = 0
- A vertical line has an undefined slope, since it would result in a fraction with 0 as the denominator. Refer to the equation provided below.

Slope is essentially the change in height over the change in horizontal distance, and is often referred to as "rise over run." It has applications in gradients in geography as well as civil engineering, such as the building of roads. In the case of a road, the "rise" is the change in altitude, while the "run" is the difference in distance between two fixed points, as long as the distance for the measurement is not large enough that the earth's curvature should be considered as a factor. The slope is represented mathematically as:

In the equation above, y 2 - y 1 = Δy , or vertical change, while x 2 - x 1 = Δx , or horizontal change, as shown in the graph provided. It can also be seen that Δx and Δy are line segments that form a right triangle with hypotenuse d , with d being the distance between the points (x 1 , y 1 ) and (x 2 , y 2 ) . Since Δx and Δy form a right triangle, it is possible to calculate d using the Pythagorean theorem. Refer to the Triangle Calculator for more detail on the Pythagorean theorem as well as how to calculate the angle of incline θ provided in the calculator above. Briefly:

d = √ (x 2 - x 1 ) 2 + (y 2 - y 1 ) 2

The above equation is the Pythagorean theorem at its root, where the hypotenuse d has already been solved for, and the other two sides of the triangle are determined by subtracting the two x and y values given by two points. Given two points, it is possible to find θ using the following equation:

m = tan(θ)

Given the points (3,4) and (6,8) find the slope of the line, the distance between the two points, and the angle of incline:

d = √ (6 - 3) 2 + (8 - 4) 2 = 5

While this is beyond the scope of this calculator, aside from its basic linear use, the concept of a slope is important in differential calculus. For non-linear functions, the rate of change of a curve varies, and the derivative of a function at a given point is the rate of change of the function, represented by the slope of the line tangent to the curve at that point.

## Slope Formula Calculator

Enter any Number into this free calculator.

$ \text{Slope } = \frac{ y_2 - y_1 } { x_2 - x_1 } $

How it works: Just type numbers into the boxes below and the calculator will automatically find the slope of two points

How to enter numbers: Enter any integer, decimal or fraction . Fractions should be entered with a forward slash such as '3/4' for the fraction $$ \frac{3}{4} $$ .

(x 1 , y 1 )

(x 2 , y 2 )

- All calculations

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## Slope Calculator

Find the slope of a line step by step.

The calculator will find the slope of the line passing through the two given points or the slope of the given line, with steps shown.

Related calculators: Line Calculator , Parallel and Perpendicular Line Calculator

Choose type: Two points Line

Enter two points or

Point 1: ( , )

Point 2: ( , )

If the calculator did not compute something or you have identified an error, or you have a suggestion/feedback, please write it in the comments below.

Our Slope Calculator is your primary resource for effortlessly grasping and determining slopes. Beyond just calculating the slope between two given points, it also offers a step-by-step solution.

## How to Use the Slope Calculator?

Locate the input fields for your two points. Enter the $$$ x $$$ and $$$ y $$$ coordinates of the first point. Similarly, input the $$$ x $$$ and $$$ y $$$ coordinates of the second point. Alternatively, enter the equation of the given line.

## Calculation

Once you've entered the data, click the "Calculate" button.

The calculated slope will be displayed in the results section.

## How to Find the Slope of a Line?

The slope of a line, often denoted by the letter $$$ m $$$ , measures how steep the line is. It describes how much the line rises (or falls) for a horizontal movement. In more technical terms, the slope represents the rate of change of the y-values compared to the x-values.

## How Do You Determine the Slope of a Line?

To determine the slope of a line, follow these steps:

- Identify Two Points on the Line: Choose two distinct points on the line, preferably ones that are easy to work with. Label them as $$$ \left(x_1,y_1\right) $$$ and $$$ \left(x_2,y_2\right) $$$ .

Use the Slope Formula: The slope is calculated using the following formula:

This formula calculates the vertical change rate (rise) to the horizontal change (run) rate between the two points.

- For a horizontal line, which doesn't rise or fall, the slope is $$$ 0 $$$ .
- The slope is undefined for a vertical line, and it doesn't run horizontally because dividing by zero (the horizontal change) is impossible.

## How to Interpret the Slope

Even though the slope of a line doesn't reveal its placement on the graph, it does describe the line's tilt or angle.

- When the slope is positive, the line is inclined upwards when observed on the graph from left to right.
- Conversely, a negative slope means the line falls from left to right.
- A slope of zero results in a horizontal line, showing neither an upward nor downward inclination. Such a line is often referred to as having no slope.
- For vertical lines, the slope cannot be defined. This is due to the constant x-value for any two points on the line, resulting in a zero difference. Consequently, determining the slope would mean dividing by zero, which means that the slope is either infinite or undefined.

## How to Find the x- and y-Intercepts of a Line

A line's x- and y-intercepts are the points where the line crosses the x-axis and y-axis, respectively. Knowing how to find these intercepts can provide valuable insights into the properties of the line. Here's a step-by-step guide for determining them:

Finding the x-intercept:

- Set $$$ y=0 $$$ in the equation of the line.
- Solve for $$$ x $$$ . The solution will give the x-coordinate of the x-intercept.
- The x-intercept will be of the form $$$ (x,0) $$$ .

Finding the y-intercept:

- Set $$$ x=0 $$$ in the equation of the line.
- Solve for $$$ y $$$ . The solution will give the y-coordinate of the y-intercept.
- The y-intercept will be of the form $$$ (0, y) $$$ .

Understanding how to find the x and y intercepts is foundational in algebra. It aids in graphing linear equations and understanding the behavior of lines within the coordinate plane. It also helps to find the slope of a line because if $$$ (x_0,0) $$$ and $$$ (0,y_0) $$$ are the two intercepts of the line, its slope is $$$ m=-\frac{y_0}{x_0} $$$ .

## Why Choose Our Slope Calculator?

Our calculator is designed with precision in mind. It ensures you get correct results every time, eliminating the risk of manual calculation errors.

## User-Friendly Interface

With its intuitive design, users of all levels can effortlessly input their data and get results. No more complicated menus or instructions.

In seconds, you can input your coordinates and receive the calculated slope. This efficiency saves time, especially for those working on multiple problems or real-world projects.

## Versatility

Our calculator supports different inputs: a line or two points.

## How is finding the slope used in real life?

Finding the slope is not just a mathematical exercise; it has numerous real-life applications. The slope can help us understand the economic relationship between supply and demand. Engineers use slopes to determine the gradient of roads or railways. In environmental science, it can indicate the steepness of a terrain which impacts drainage, erosion, and construction. Financial analysts use it to see the trend of stock prices over time. In essence, in any situation where there's a need to understand the rate of change or relationship between two variables, the concept of a slope comes into play.

## How do you find the slope of a curve?

Curves don't have a consistent slope like straight lines do. The slope at a particular point on a curve is the slope of its tangent at that point. Using calculus, specifically differentiation, the slope of the function $$$ f(x) $$$ at some point $$$ x_0 $$$ is given by its derivative $$$ f'(x_0) $$$ . By evaluating the derivative, you can find the curve's slope.

## What is the primary function of the Slope Calculator?

The Slope Calculator is designed to compute the slope of the given line or the line through the two given points.

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- Slope Ca...

## Slope Calculator

Choose your calculator☝️

Enter the coordinates ( x 1 , y 1 ) \hspace{0.2em} (x_1, y_1) \hspace{0.2em} ( x 1 , y 1 )

Enter the coordinates ( x 2 , y 2 ) \hspace{0.2em} (x_2, y_2) \hspace{0.2em} ( x 2 , y 2 )

Hello there!

## About the Slope Calculator

This slope calculator lets you calculate the slope of a line if you know any two istinct points on the line or the the line's equation.

The calculator will tell you not only the slope, but also how to calculate it.

## Usage Guide

I. valid inputs.

Each input can be a real number in any format — integers, decimals, fractions, or even mixed numbers. Here are a few examples.

- Whole numbers or decimals → 2 \hspace{0.2em} 2 \hspace{0.2em} 2 , − 4.25 \hspace{0.2em} -4.25 \hspace{0.2em} − 4.25 , 0 \hspace{0.2em} 0 \hspace{0.2em} 0 , 0.33 \hspace{0.2em} 0.33 \hspace{0.2em} 0.33
- Fractions → 2 / 3 \hspace{0.2em} 2/3 \hspace{0.2em} 2/3 , − 1 / 5 \hspace{0.2em} -1/5 \hspace{0.2em} − 1/5
- Mixed numbers → 5 1 / 4 \hspace{0.2em} 5 \hspace{0.5em} 1/4 \hspace{0.2em} 5 1/4

IMPORTANT — When providing inputs for the linear equation, the coefficients of x \hspace{0.2em} x \hspace{0.2em} x and y \hspace{0.2em} y \hspace{0.2em} y cannot both be 0 \hspace{0.2em} 0 \hspace{0.2em} 0 at the same time. That would eliminate both variables from the equation and hence it will no longer be a linear equation.

## ii. Example

If you would like to see an example of the calculator's working, just click the "example" button.

## iii. Solutions

As mentioned earlier, the calculator won't just tell you the answer but also the steps you can follow to do the calculation yourself. The "show/hide solution" button would be available to you after the calculator has processed your input.

We would love to see you share our calculators with your family, friends, or anyone else who might find it useful.

By checking the "include calculation" checkbox, you can share your calculation as well.

Here's a quick overview of the slope of a line.

For those interested, we have a more comprehensive tutorial on the slope of a line and its calculation .

## Slope of a Line

In analytical geometry, we often need to find the slope between two points or two parallel lines or from a point to a line.

The slope of a line is a measure of its steepness and direction. It is the ratio of the rise (change in y \hspace{0.2em} y y –coordinate) to the run (change in x \hspace{0.2em} x x –coordinate) as we move from one point to another on the line.

## Finding the Slope

How we find the slope depends on what information we have about the line. Let's look at a couple of important cases.

## Two Points on the Line

If a line passes through points ( x 1 , y 0 ) \hspace{0.2em} (x_1, \hspace{0.2em} y_0) \hspace{0.2em} ( x 1 , y 0 ) and ( x 2 , y 2 ) \hspace{0.2em} (x_2, \hspace{0.2em} y_2) \hspace{0.2em} ( x 2 , y 2 ) , its slope m \hspace{0.2em} m \hspace{0.2em} m is given by the formula —

## Equation of the Line

If the equation of a line is of the form y = m x + b \hspace{0.2em} y = {\color{Red} m} x + b \hspace{0.2em} y = m x + b (slope–interecept form), it's slope would be m \hspace{0.2em} {\color{Red} m} \hspace{0.2em} m .

If the equation of the line is in some other form, we first convert it into the slope– intercept form, them extract the slope from it.

## Slope Calculation

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## Slope Calculator

How to use the slope calculator.

Slope Calculator is an easy-to-use tool. You will have to

- Choose the method to find the slope of line.
- In case of two points, enter X 1 , Y 1 , X 2 , and Y 2 .
- If you have chosen the Line Equation method, enter the coefficients.
- Click Calculate to get the slope of line.

Line Equation

Formula to find

## Slope (m) =

What is slope calculator.

The slope formula calculator has great importance in both Mathematics and Physics. It helps to find the gradient (slope) of a line by taking two points or line equations as input.

In addition to finding the simple slope, it finds a whole lot of other slope and line characteristics as well. These include:

- Slope intercept form
- Percentage grade, distance, and angle
- Graph of the slope

## What is the slope of a line?

Slope in math has the same meaning as in English and that is “ Steepness ”. Except in maths, we use the word slope for the steepness of lines and curves.

It is also known as gradient, incline, and slant. Daily life examples of slope include roofs, slides, and steep mountains.

## Slope Formula

The slope formula is ‘rise over run’. Mathematically it can be written as:

m = (Y 2 - Y 1 ) / (X 2 - X 1 )

m = 𝚫Y / 𝚫X

These can be any two-point of a line in a cartesian plane.

## Types of slope

Slope can be positive, negative, zero, or undefined. The types of slope depend on their values and the sign with the value. For a quick review, a slope that has

- A positive sign then is positive
- A negative sign then is of course negative
- Horizontal value only means it is zero
- Vertical value only indicates it is undefined.

## How to find the Slope of a line?

If you need to find a slope quickly and without any error, you can use the slope finder for that purpose. But if you want to calculate it yourself, keep on reading the example below.

## 1. Slope Using Two Points

Find the slope of the line passing through points (3,6) and (8,2).

Step 1: Identify the values.

X 2 = 8

Y 1 = 6

Y 2 = 2

Step 2: Find the difference between the points.

𝚫X = X 2 - X 1

𝚫Y = Y 2 - Y 1

Step 3: Solve the fraction 𝚫Y / 𝚫X.

= -4 / 5

Hence the slope of the line is -0.8 and negative in nature.

## 2. Slope using Line Equation

Slope can also be found if you have an equation of a line. Let’s find the slope using the line equation.

Find the slope of line in the following line equation.

4y – 2x + 5 = 0

Step 1: Arrange the equation in the form of y = mx + c

4y = 2x – 5

y = (2x – 5)/4

Step 2: Simplify the right side of the equation.

y = 2x/4 – 5/4

y = 0.5x – 1.25

Now that we have the equation of the straight line, a slope can be found by comparing it to the original equation where m represents the slope.

y = mx + c

y = 0.5x – 1.25

Slope (m) = 0.5

## Frequently Asked Question

What is meant by the slope of a line?

The slope is the measure of the rise over run of the given coordinate points of x and y. Mathematically,

Rise over run is (y2 - y1)/(x2 - x1)

What is the symbol of the slope?

The measure of the steepness of the line is denoted by the letter "m ". Such as

m = rise/run

- Khan Academy. (n.d.). What is the slope of a line? Khan Academy.
- Learning, L. (n.d.). How to find the Slope of a line? Lumen.

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## Circumference of a Circle: Definition, Formula, Properties, and Examples

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## Slope Intercept Form: Definition, Properties, and Examples

## Slope Calculator with steps

Instructions: Use this calculator to get the slope of a line, with all the calculations shown, step-by-step. In order to do so, you need to provide indicate the line for which you need to compute the slope.

Please provide either a valid linear equation or two points \((x_1, y_1)\) and \((x_2, y_2)\) where the line passes through.

Select one of the options

## How to Use this slope calculator with steps

Use this calculator to find the slope of a line that you provide, with all the steps shown.

The slope is a measure of the inclination of the line with respect to the coordinated axes. A positive slope indicates that the line has a upward inclination, whereas a negative slope indicates that the line has a downward inclination.

A slope equal to zero indicates that the line is horizonal, whereas a vertical line does not have a well defined slope.

## How to calculate a slope?

Normally the calculation of the slope is easy, but there are multiple ways a slope can be calculated, and it all depends on what information is provided and how it is provided.

The most common ways that a slope calculation is presented is by first providing you with a linear equation for which you need to find its slope, or when they provide you with two points a line passes through.

## Slope Calculator from Equation: Finding the slope of the line

This calculator will show you how to compute the slope of a line you provide, and you will have different ways to indicate and define your line. It also will give you a graph reflecting the slope calculated.

For example, one common way is to define your line giving an equation, and then you will have this calculator to compute the slope from the equation.

The general strategy for that is to put the equation of the line in slope-intercept form , from which point is easy to recognize the slope from the structure of the equation \(y = mx + n\).

## This is a slope from two points calculator too

Perhaps one of the most common ways to compute the slope is when you define the equation by providing two points as \((x_1, y_1)\), \((x_2, y_2)\). So, how do you find the slope from two points? Te slope is simply computed as

which is how to find the slope from two points. Let's not forget that those two points are usually points where a line passes through, so you are finding the slope of the line that passes through those points.

Ultimately, how to find the slope of a line will depend on how the line is defined. This calculator will get you covered with all the cases, even when there are fractions in the calculation.

## Interpretation: What is a slope of 2%?

There are several ways of seeing this, but a common way is to think that for each increase in 100 units in X, the line increases 2 units in Y, which explains the 2/100 = 0.02 = 2%.

Along the same line of interpretation, you can say that a slope of 45% is such that an increase of 100 units in X leads to an increase of 45 units in Y. Please notice that this is NOT the same as a slope with 45 o degrees.

## Instant Slope Calculator

The idea of a slope calculator is simple when you consider two points, in which case you use the formula above. But what is the instant slope? That refers to the slope when the two points become increasing close.

So you want to see what value the slope approaches too, when the two points approach together. The idea of instant slope is reflected by this derivative calculator , which is essentially computing instant slopes.

## Example: Calculation of the slope

Suppose that you have a line that has the following standard form \( \frac{3}{4} x + 2y = 6\). Find slope of the line.

We have been provided with the following equation:

Putting \(y\) on the left hand side and \(x\) and the constant on the right hand side we get

Now, solving for \(y\), by dividing both sides of the equation by \(2\), the following is obtained

and simplifying we finally get the following

Conclusion : Based on the data provided, we conclude that the slope of the line is \(\displaystyle m = -\frac{3}{8}\).

## Example: Calculation of Slope from two points

Suppose that you have a line that has passes through 2 points: \( (1, 2)\) and \( (4, 11/3)\). Find slope of the line.

## Calculation of the Slope of a line

The information provided about the line is that the line passes through the points\(\displaystyle \left( 1, 2\right)\) and \(\displaystyle \left( 4, \frac{11}{3}\right)\)

So then, we find that the slope is \(\displaystyle m = \frac{5}{9}\) and that the line passes through the point \(\displaystyle \left( 1, 2\right)\)

Conclusion : Based on the data provided, we conclude that the slope of the line is \(\displaystyle m = \frac{5}{9}\).

The slope of a line is one of its most important properties, along with the y-intercept and x-intercept , because they essentially define the line.

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## Statistic & Probability

Trigonometric.

Slope is basically the amount of slant a line has, and can have a positive, negative, zero or undefined value.

## Slope Formula

slope = y 2 -y 1 x 2 -x 1

## Slope Intercept Form

y = mx + b (with m represents the slope and b represents the intercept ) is the relationship of a straight line.

What is the slope between (0, 5) and (7, 0)?

slope = 0 - 5 7 - 0 = - 5 7

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## Slope Intercept Form Calculator

What is the slope intercept form, slope intercept formula derivation, how to find the equation of a line, find the x-intercept and y-intercept, real world uses of y-intercept and x-intercept, other equations with y-intercept, equations with no intercept (asymptote), intercepts and linear equations in machine learning and science.

The slope intercept form calculator will teach you how to find the equation of a line from any two points that this line passes through. It will help you to find the coefficients of slope and y-intercept, as well as the x-intercept, using the slope intercept formulas. Read on to learn what is the slope intercept form of a linear equation, how to find the equation of a line and the importance of the slope intercept form equation in real life.

Any line on a flat plane can be described mathematically as a relationship between the vertical (y-axis) and horizontal (x-axis) positions of each of the points that contribute to the line. This relation can be written as y = [something with x] . The specific form of [something with x] will determine what kind of line we have. For example, y = x² + x is a parabola, also called a quadratic function. On the other hand, y = mx + b (with m and b representing any real numbers) is the relationship of a straight line.

In this slope intercept calculator, we will focus only on the straight line . You can check our average rate of change calculator to find the relation between the variables of non-linear functions.

Linear equations, or straight-line equations, can be quickly recognized as they have no terms with exponents in them . (For example, you will find an x or a y , but never an x² .) Each linear equation describes a straight line, which can be expressed using the slope intercept form equation.

As we have seen before, you can write the equation of any line in the form of y = mx + b . This is the so-called slope intercept form because it gives you two important pieces of information: the slope m and the y-intercept b of the line. You can use these values for linear interpolation later.

The term slope is the incline, or gradient, of a line. It tells us how much y changes for a fixed change in x . If it is positive, the values of y increase when x increases. If it is negative, y decreases with an increasing x . You can read more about it in the description of our slope calculator .

The y-intercept is the value of y at which the line crosses the y-axis. To find it, you have to substitute x = 0 in the linear equation. You will see later why the y-intercept is an important parameter in linear equations , and you will also learn about the physical meaning of its value in certain real-world examples.

Still need to know how to find the slope intercept form of a linear equation? We will assume you know two points that the straight line goes through. The first one will have coordinates (x₁, y₁) and the second one (x₂, y₂). Your unknowns are the slope m and the y-intercept b .

Firstly, substitute the coordinates of the two points into the slope intercept equation:

(1) y₁ = mx₁ + b

(2) y₂ = mx₂ + b

Then, subtract the first equation from the second:

y₂ - y₁ = m(x₂ - x₁)

Finally, divide both sides of the equation by (x₂ - x₁) to find the slope:

m = (y₂ - y₁)/(x₂ - x₁)

Once you have found the slope, you can substitute it into the first or second equation to find the y-intercept:

y₁ = x₁(y₂ - y₁)/(x₂ - x₁) + b

b = y₁ - x₁(y₂ - y₁)/(x₂ - x₁)

This slope intercept form calculator allows you to find the equation of a line in the slope intercept form . All you have to do is give two points that the line goes through. You need to follow the procedure outlined below.

Write down the coordinates of the first point . Let's assume it is a point with x₁ = 1 and y₁ = 1 .

Write down the coordinates of the second point as well. Let's take a point with x₂ = 2 and y₂ = 3 .

Use the slope intercept formula to find the slope:

m = (y₂ - y₁)/(x₂ - x₁) = (3-1)/(2-1) = 2/1 = 2 .

Calculate the y-intercept . You can also use x₂ and y₂ instead of x₁ and y₁ here.

b = y₁ - m × x₁ = 1 - 2×1 = -1

Put all these values together to construct the slope intercept form of a linear equation:

y = 2x - 1 .

You can also use the distance calculator to find the distance between two points.

It is also always possible to find the x-intercept of a line . It is the value of x at which the straight line crosses the x-axis (it means the value of x for which y equals 0 ). You can calculate it in the following way:

As we can see, the only condition that must be met is that coefficient m is different from zero.

Our slope intercept form calculator will display both the values of the x-intercept and y-intercept for you. Still, if you would like to learn more about them, we recommend you visit our x- and y-intercept calculator .

We have already seen what is the slope intercept form, but to understand why the slope intercept form equation is so useful, you should know some applications it has in the real world . Let's see a couple of examples. We will start with simple ones from physics so that you can get an intuitive idea of what the y-intercept and x-intercept mean.

Imagine a car moving at a fixed speed toward you . Its movement can be plotted as time versus the distance the car is from you (as shown above). This means that the x-axis will represent the time passed, and the y-axis will represent the distance to the car. You can even imagine the car has started to move before you started the timer (that is: before t = 0 ).

Now, if you look at the y-intercept ( x = 0 ), the point at which you started to keep track of time is t = 0 . And so, the value of y at this point will indicate the starting position (distance) of the car with respect to you. This value is, as we have discussed before, the same as the value of b in the slope intercept form of a straight-line equation.

Looking now at the x-intercept ( y = 0 ), this will be the point at which the distance from the car to you will be 0 . Then the value of x at this point will be the time when you and the car were at the same place . Let's hope that means you were inside the car and not under.

The car example above is a very simple one that should help you understand why the slope intercept form is important and, more specifically, the meaning of the intercepts . In this article, we will mostly talk about straight lines, but the intercept points can be calculated for any kind of curve (if it crosses an axis).

In fact, the example above does not fit a linear equation and still has both intercepts. The same is true for any other parabola or another shape.

One equation that is guaranteed to have a y-intercept but not necessarily an x-intercept is a parabola. This is equation is shown in the image above. It has a maximum or a minimum (depending on the orientation). If this maximum is below the x-axis or the minimum is above the x-axis, there will never be an x-intercept .

However, unlike humans, not all equations are equal . Some of the formulas describe curves that might never intercept the x-axis, the y-axis, or both. Let's see in a bit more detail how this can be.

We can distinguish 3 groups of equations depending on whether they have a y-intercept only, an x-intercept only, or neither . The first group (y-intercept only) can have almost any type of equation, including linear equations. A good easy example is y = 3 (or any other constant value of y except for 0 ) since this is a line parallel to the x-axis and will, thus, never cross or intercept it. Please don't try to calculate these types of intercepts on this slope intercept form calculator as these types of equations can potentially break the Internet .

The second and third groups of equations are a bit more tricky to imagine and to understand them well, we need to introduce the concept of an asymptote . An asymptote is a line (that can be expressed as a linear equation) to which the function or curve we are talking about gets closer and closer but never actually crosses or touches that line.

The definition might not seem totally clear, but if we look at an example equation , we will have fewer problems with understanding it. Let's take the equation y = 1/x . If we try to find the y-intercept by substituting x = 0 , we arrive at what is called a mathematically undefined expression since it makes no sense to divide by 0 .

If we take values closer and closer to 0 (something like 0.1 , then 0.001 , 0.000001 ...), we can see that the value of y increases very rapidly. So around the point x = 0 , we know that y would have a massive value, but because of how math works, it does not have a defined value for that exact point. Sometimes people may say 1/0 = ∞ , but the reality is that infinity is not a number but a concept .

In this case, the linear equation x = 0 represents the asymptote of the function y = 1/x , which means that y = 1/x will never intercept that line and, thus, will not have a y-intercept. In general, any time that a function has an asymptote that lies on one of the axes, it will be missing at least one of the intercepting points .

In fact, the example we have shown you ( y = 1/x ) also has an asymptote for y = 0 , i.e., the x-axis. For the same reason as before, y = 0 is never achievable by the formula because it would require x = ∞ , and as we said before, it is impossible to achieve that since infinity is a concept and not a number .

Before we move to our next topic, it is important to note that we have made extreme over-simplifications when talking about infinity, but we feel it is a good and fast approach for those that are not used to the concept of working with infinity in math. We recommend that you learn more about the proper ways of infinity , starting with the undefined expressions in math .

One could easily think that the usefulness of linear equations is very limited due to their simplicity. However, the reality is a bit different. Linear equations are at the core of some of the most powerful methods to solve minimization and optimization problems .

Minimization problems are a type of problem in which one would like to find how to make one of the variables as small as possible. This variable could be, for example, the difference between a prediction made by a model and reality. These types of problems are one of the most common problems and are at the core of machine learning and scientific experiments.

One of the most common and powerful methods to find the minimum value of an equation or formula is the so-called Newton method , named after the genius that invented it. The way it works is by using derivatives, linear equations, and x-intercepts:

This method consists of choosing a value of x for the equation and calculating the derivative of the equation at that point. Using the derivative as the slope of a linear equation that passes through that exact (x, y) point, the x-intercept is then calculated. This is one of the situations in which the slope intercept form comes in handy.

Once the x-intercept is calculated, that value of x is used to repeat the process above, a specific number of times , until we arrive at a value of y that is minimum (which means that the derivative will be 0 ). In real life, arriving at the exact minimal point is not possible to do in a finite amount of time, so typically, people will settle for a "close enough" value.

One very common example is when using the least squares method to fit some data to a formula or trend. In this case, the value that we want to minimize is the sum of the squared distance from the trend line to the data points, where the distance is calculated along a perpendicular line from the point to the trend line.

## Is slope intercept form the same as standard form?

No , standard form, and slope-intercept form are two different ways of describing a line:

- Slope intercept form reads y = mx + b , where m is the slope (steepness) of the line, and b is the y-intercept , i.e., the value at which the line intersects the vertical axis. For example, y = -2x + 3 .
- Standard form reads Ax + By + C = 0 , where A, B, C are integers . For example, 2x + y - 3 = 0 .

## How do I convert standard form to slope intercept form?

If you want to rewrite your standard form equation to the slope-intercept form, follow these steps:

- Write down the standard form of your line: Ax + By + C = 0 .
- Move Ax and C on the right-hand side so that By remains by itself on the left side: By = -Ax - C .
- Divide both sides by B : y = -(A/B)x - (C/B) .
- As you can see, we've got the slope intercept form y = mx + b with slope m = -A/B and intercept b = -C/B .

## How do I interpret the slope of a line?

The slope (aka gradient ) describes the steepness of a line. Slope can be positive, negative, or zero and:

- Positive slope means the line rises from left to right.
- Negative slope means the line goes downwards from left to right.
- Zero gradient means the line is horizontal.

The bigger the absolute value of the gradient, the faster the line increases/decreases. In fact, the value of the slope is exactly the amount by which the line increases/decreases when x increases by one unit.

## What is the slope of a line inclined at angle 45°?

The slope is m = 1 . To get this result, use the formula ‘m = tan(α)’, where α is the angle between the line and the x-axis. Since tan(45°) = 1 , we get the slope 1 , as claimed.

## Rectangular pyramid volume

Supplementary angles.

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## Slope Calculator

If we talk about the angle of slope which can be worked out using the Slope Calculator, then we can mean a range of various things whose angle of the slope can found. For example, the angle of the driveway, the pitch of a roof, the angle of a hill, etc.

Carrying out the slope calculations can be very helpful in different situations that range from making sure that the water flow runs exactly off a particular surface. Also, tension, friction and energy can be reduced as well, if some heavy objects are moved using a ramp.

It is important to understand what is a slope. A Slope can be referred to as a piece of ground that has a specific slant, which is also called a "Grade".

There are 4 different ways that can be used to specify the slope: the angle of inclination, percentage , per mille, and ratio.

Note: The ratio is determined to be "1 in n" which is different to the mathematical "1:n" .

The Gradient Calculator is another online tool that may be useful to you.

Angle of Inclination:

Slope Length:

You may set the number of decimal places in the online calculator. By default there are only two decimal places.

0 1 2 3 4 5 6 7 8 9 Decimal Places

## Some Basic Information About the Slope Calculator

The Slope Calculator is capable of carrying out mathematical operations with the following algorithms:

- Slope Length is the square root of (Rise squared plus Run squared)
- Angle of Inclination is the arctan of (Rise divided by Run)
- Percentage is 100 multiplied by (Rise divided by Run)
- Per Mille is 1000 multiplied by (Rise divided by Run)
- Ratio is 1 to (Run divided by Rise)
- Currently 4.26/5

Rating: 4.3 /5 (333 votes)

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## Course: Algebra 1 > Unit 4

- Intro to slope
- Positive & negative slope
- Worked example: slope from graph
- Slope from graph
- Graphing a line given point and slope
- Graphing from slope
- Calculating slope from tables
- Slope in a table
- Worked example: slope from two points
- Slope from two points

## Slope review

What is slope, example: slope from graph, example: slope from two points.

- Your answer should be
- an integer, like 6
- a simplified proper fraction, like 3 / 5
- a simplified improper fraction, like 7 / 4
- a mixed number, like 1 3 / 4
- an exact decimal, like 0.75
- a multiple of pi, like 12 pi or 2 / 3 pi

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## Slope (Gradient) of a Straight Line

The Slope (also called Gradient ) of a line shows how steep it is.

To calculate the Slope:

Have a play (drag the points):

## Positive or Negative?

Going from left-to-right, the cyclist has to P ush on a P ositive Slope:

When measuring the line:

- Starting from the left and going across to the right is positive (but going across to the left is negative).
- Up is positive , and down is negative

That line goes down as you move along, so it has a negative Slope.

## Straight Across

A line that goes straight across (Horizontal) has a Slope of zero.

## Straight Up and Down

That last one is a bit tricky ... you can't divide by zero , so a "straight up and down" (vertical) line's Slope is "undefined".

## Rise and Run

Sometimes the horizontal change is called "run", and the vertical change is called "rise" or "fall":

They are just different words, none of the calculations change.

## Math Calculator

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Free slope calculator - find the slope of a line given two points, a function or the intercept step-by-step

The slope calculator determines the slope or gradient between two points in the Cartesian coordinate system. The slope is basically the amount of slant a line has and can have a positive, negative, zero, or undefined value. Before using the calculator, it is probably worth learning how to find the slope using the slope formula.

Slope Calculator Slope Calculator - Two Points use whole numbers, fractions or decimals ( x1 y1 ) = ( x2 y2 ) = Answer: Slope m = − 12 5 As a decimal: m = -2.4 Graph of the line y = mx + b 1 2 3 4 5 6 7 8 9 −1 −2 2 4 −2 −4 −6 −8 −10 0,0 - o + ← ↓ ↑ → X Y (2, 3) (7, -9) Graph tools will zoom, move and enter full screen mode.

Briefly: d = √ (x2 - x1)2 + (y2 - y1)2 The above equation is the Pythagorean theorem at its root, where the hypotenuse d has already been solved for, and the other two sides of the triangle are determined by subtracting the two x and y values given by two points. Given two points, it is possible to find θ using the following equation: m = tan (θ)

Algebra Slope and Y-Intercept Calculator Step 1: Enter the linear equation you want to find the slope and y-intercept for into the editor. The slope and y-intercept calculator takes a linear equation and allows you to calculate the slope and y-intercept for the equation.

Step 1: Enter the point and slope that you want to find the equation for into the editor. The equation point slope calculator will find an equation in either slope intercept form or point slope form when given a point and a slope. The calculator also has the ability to provide step by step solutions. Step 2: Click the blue arrow to submit.

Enter any Number into this free calculator. Slope = y2 −y1 x2 −x1 Slope = y 2 − y 1 x 2 − x 1. How it works: Just type numbers into the boxes below and the calculator will automatically find the slope of two points. How to enter numbers: Enter any integer, decimal or fraction. Fractions should be entered with a forward slash such as '3/ ...

To determine the slope of a line, follow these steps: Identify Two Points on the Line: Choose two distinct points on the line, preferably ones that are easy to work with. Label them as (x1,y1) ( x 1, y 1) and (x2,y2) ( x 2, y 2). Use the Slope Formula: The slope is calculated using the following formula: m = y2 −y1 x2 −x1 m = y 2 − y 1 x ...

In analytical geometry, we often need to find the slope between two points or two parallel lines or from a point to a line. The slope of a line is a measure of its steepness and direction. It is the ratio of the rise (change in \hspace {0.2em} y y -coordinate) to the run (change in \hspace {0.2em} x x -coordinate) as we move from one point ...

What is Slope Calculator? The slope formula calculator has great importance in both Mathematics and Physics. It helps to find the gradient (slope) of a line by taking two points or line equations as input. In addition to finding the simple slope, it finds a whole lot of other slope and line characteristics as well. These include: Slope ...

Solvers Algebra Slope Calculator with steps Instructions: Use this calculator to get the slope of a line, with all the calculations shown, step-by-step. In order to do so, you need to provide indicate the line for which you need to compute the slope.

Free math problem solver answers your algebra homework questions with step-by-step explanations.

Calculator slope = y2-y1 x2-x1 Point 1 ( , ) Point 2 ( , ) Slope = Intercept form Knowledge Slope Slope is basically the amount of slant a line has, and can have a positive, negative, zero or undefined value. Slope Formula slope = y2-y1 x2-x1 Slope Intercept Form

Symbolab, Making Math Simpler. Word Problems. Provide step-by-step solutions to math word problems. Graphing. Plot and analyze functions and equations with detailed steps. Geometry. Solve geometry problems, proofs, and draw geometric shapes. Math Help Tailored For You.

The slope intercept form calculator will teach you how to find the equation of a line from any two points that this line passes through. It will help you to find the coefficients of slope and y-intercept, as well as the x-intercept, using the slope intercept formulas.

The Slope Calculator is capable of carrying out mathematical operations with the following algorithms: Slope Length is the square root of (Rise squared plus Run squared) The angle of a slope can be calculated using the online Slope calculator. Examples of the angle of a slope include such things as the angle of the driveway, the pitch of a roof ...

Get math help in your language. Works in Spanish, Hindi, German, and more. Online math solver with free step by step solutions to algebra, calculus, and other math problems. Get help on the web or with our math app.

Symbolab is the best step by step calculator for a wide range of physics problems, including mechanics, electricity and magnetism, and thermodynamics. It shows you the steps and explanations for each problem, so you can learn as you go. How to solve math problems step-by-step?

Step 1: Identify the values of x 1 , x 2 , y 1 , and y 2 . x 1 = 2 y 1 = 1 x 2 = 4 y 2 = 7 [Explain] Step 2: Plug in these values to the slope formula to find the slope. Slope = y 2 − y 1 x 2 − x 1 = 7 − 1 4 − 2 = 6 2 = 3 Step 3: Gut check. Make sure this slope makes sense by thinking about the points on the coordinate plane.

So the slope (m) is -1/2. The main thing to keep track of is which point is (x₁, y₁) and which point is (x₂, y₂). You don't want to mix these up. A few tips for graphs of slopes: - a perfectly horizontal line has no slope - a perfectly vertical line has a slope that is not defined - a line that goes upwards (from left to right) has a ...

The Slope of this line = 3 3 = 1. So the Slope is equal to 1. The Slope of this line = 4 2 = 2. The line is steeper, and so the Slope is larger. The Slope of this line = 3 5 = 0.6. The line is less steep, and so the Slope is smaller.

The Math Calculator will evaluate your problem down to a final solution. You can also add, subtraction, multiply, and divide and complete any arithmetic you need. Step 2: Click the blue arrow to submit and see your result! Math Calculator from Mathway will evaluate various math problems from basic arithmetic to advanced trigonometric expressions.

Explore math with our beautiful, free online graphing calculator. Graph functions, plot points, visualize algebraic equations, add sliders, animate graphs, and more.