Slope of a Linear Equation
Support Page

Welcome to our Slope of a Linear Equation Support Page.

Looking for some help and support for finding the slope from a linear equation?

Then check out this page with clear explanation, worked examples and some practice worksheets!

We also have a linear equation calculator which you can use to solve linear equations with two points and find out what the slope is, step-by-step!

  • This page contains links to other Math webpages where you will find a range of activities and resources.
  • If you can't find what you are looking for, try searching the site using the Google search box at the top of each page.

Slope of a Linear Equation Support

linear equation with two points example 1 image

This page contains all the information you need to find the slope of a line which goes through two (or more) points.

Find out about different values the slope can have and a quick way of finding the slope from a linear equation.

We have also included some practice worksheets and worked examples to help you to understand and learn this skill.

In order to master this skill, you should have a basic level of algebra and understand how to add and subtract negative numbers.

Quicklinks to ...

What is a Linear Equation?

A linear equation (or line equation) is an equation of a straight line.

Every straight line can be written as a linear equation.

linear equation with two points image

There are two main forms which are used to show linear equations:

Standard Form

linear equation standard form image

A linear equation can be written in the standard form:

Ax + By + C = 0

  • where A, B and C are whole numbers
  • x and y are variables

Slope-Intercept Form

linear equation slope-intercept form image

However, it is more commonly written in slope-intercept form:

y = mx + c

  • m is the gradient or slope of the line
  • x is the x coordinate and y is the y coordinate
  • the point c is the y intercept where the line crosses the y-axis.

If we are interested in finding the slope of a linear equation, we need to find the value of m.

Looking for more support with linear equations...
Try our linear equation support page

What is the Slope of a Linear Equation?

The slope of a line is also know as the gradient.

It is also the value of the letter m in the equation:

y = mx + c

The slope of a line can be expressed as a number as follows:

  • it can be a positive number which means the slope goes upwards from left to right
  • it can be a negative number which means the slope goes downwards from left to right
  • it can be a fraction or decimal
  • the higher the absolute value of the slope, the steeper it is
  • the lower the absolute value of the slope, the shallower it is
  • the slope can also be zero or infinity

There is some more information about the types of slope and their values below.

There are several ways that we can find the slope (or gradient) of a linear equation:

  • we can look at the equation itself which will show us the slope
  • we can look at two points on the linear equation to find the slope

There is more information and examples below to show you how to find the value of the slope of a linear equation.

Types of Slope from Linear Equations

If we look at a linear equation in slope-intercept form:

y = mx + c

  • where m is the slope or gradient
  • c is a constant

There are 4 main different types of slope or gradient that we might find in our straight line:

  1. line sloping upwards
  2. line sloping downwards
  3. horizontal line
  4. vertical line

Line Sloping Upwards

The value of m (the gradient or slope) is positive (m>0).

If the line slopes upwards from left to right it means that the gradient is a positive number.

linear equations positive gradient image

Examples of linear equations where the line slopes upwards:

  • y = 2x + 4
  • y = ½x - 6
  • y = 7x - 10

Line Sloping Downwards

The value of m is negative (m<0).

If the line slopes downwards from left to right then it means that the slope or gradient is negative.

linear equations negative gradient image

Examples of linear equations where the line slopes downwards:

  • y = -3x + 5
  • y = -¼x + 2
  • y = 9 - 4x
  • y = -x - 7

Horizontal Line

The value of m is zero (m=0).

If the line is a horizontal line, it means that the gradient is zero.

linear equations zero gradient image

Examples of linear equations with a horizontal slope:

  • y = 5
  • y = -2
  • y = 9
  • y = -½

Vertical Line

The value of m is infinity (m=∞).

If the line is a vertical line, it means that the slope or gradient is infinite.

linear equations infinite gradient image

Examples of linear equations with a vertical slope:

  • x = 3
  • x = -4
  • x = 9.5
  • x = -½

How to find the Slope or Gradient (m) from a Linear Equation

How to find the gradient from a linear equation

Case 1) The linear equation is in slope intercept form

This is a relatively straightforward task which does not require any working out!

If the linear equation is in the form: \[ y = mx + c \]

Then the gradient is simply the value of m.

So if the equation is: \[ y = 4x - 7 \] Then the gradient is 4.


Case 2) The linear equation is in standard form

If the linear equation is in the form: \[ Ax + By + C = 0 \]

Then we have to work a little harder to find the gradient.

Here are the simple steps you need:

Step 1) Divide all the terms by the value of B and simplify

This gives us: \[ {A \over B}x + {B \over B}y + {C \over B} = {0 \over B} \]

Simplifying gives us: \[ {A \over B}x + y + {C \over B} = 0 \]

Step 2) Move all the terms apart from the y term over to the right hand side by subtracting them

\[ y = -{A \over B}x - {C \over B} \]

The equation is now in the slope-intercept form: \[ y = mx + c \]

The value of gradient or slope: \[ m = -{A \over B} \]

So if the equation is: \[ 4x - 2y - 10 = 0 \] Then the value of A is 4 and the value of B = -2 \[ m = -{4 \over -2} = {2 \over 1} = 2 \]

So the gradient is 2.

How to find the Slope or Gradient (m) from a line on a coordinate grid

In order to do this, the easiest way is to find two points on the line which are exactly match whole numbers on the coordinate grid.

Once we have found the two points, we can use the method below for finding the slope.

How to find the Slope or Gradient (m) using two points

Formula for finding the gradient m

The gradient of the line can be found by dividing the change in the y-coordinate by the change in the x-coordinate.

We can rewrite this as a formula for m:

\[ m = {(y_2 - y_1) \over (x_2 - x_1)} \]

where the two points on the line are (x1, y1) and (x2, y2)

In other words, the slope is the difference in the value of the y-coordinates divided by the difference in value of the x-coordinates.

It does not matter which way round you input the coordinates: the slope will stay the same:

The slope between (x1, y1) and (x2, y2) has the same value as the slope between (x2, y2) and (x1, y1)

How to find the Slope Examples

Example 1) What is the gradient of the linear equation: \[ y = 3x - 8 \]

This linear equation is already in slope-intercept form (which is the form we need).

The value of m is equal to the coefficient of the x-term which is 3.

So the gradient or slope of the line is 3.

Example 2) What is the gradient of the linear equation: \[ y = 9 - 7x \]

This linear equation is also in slope-intercept form.

The value of m is equal to the coefficient of the x-term which is -7.

So the gradient or slope of the line is -7.

Example 3) Find the gradient of the linear equation: \[ 4x + y - 9 = 0 \]

This linear equation is in standard form.

We need to change this into slope-intercept form to find the gradient.

We do this by moving the x-term and the number 9 to the right hand side of the equation.

\[ 4x + y - 9 = 0 \] Subtracting 4x and adding 9 to both sides of the equation gives us: \[ y = 9 - 4x \]

The value of m is equal to the coefficient of the x-term which is -4.

So the gradient or slope of the line is -4.

Example 4) Find the gradient of the linear equation: \[ 10x - 2y + 12 = 0 \]

This linear equation is in standard form.

We need to change this into slope-intercept form to find the gradient.

First we divide each term by 2 to give: \[ 5x - y + 6 = 0 \]

Next we move the y-term to the right hand side of the equation.

\[ 5x - y + 6 = 0 \] Adding y to both sides of the equation gives us: \[ 5x + 6 = 0 + y \]

We simplify this to give us: \[ y = 5x + 6 \]

The value of m is equal to the coefficient of the x-term which is 5.

So the gradient or slope of the line is 5.

Example 5) Find the gradient of the line which passes through points: (2, -1) and (6, 11).

The equation to find the gradient is:

\[ m = {(y_2 - y_1) \over (x_2 - x_1)} \]

Our two points are (2, -1) and (6, 11) so this give us:

  • x1 = 2
  • y1 = -1
  • x2 = 6
  • y2 = 11

This gives us:

\[ m = {(11 - (-1)) \over (6 - 2)} = {12 \over 4} = 3 \]

This means that:

\[ y = 3x + c \]

As we are only looking to find the gradient, we do not need to worry about finding the value of c.

The value of m is the coefficient of the x term which is 3.

So the gradient or slope of the line is 3.

Example 6) Find the gradient of the line which passes through points: (-4, 7) and (-1, -5).

The equation to find the gradient is:

\[ m = {(y_2 - y_1) \over (x_2 - x_1)} \]

Our two points are (-4, 10) and (-1, -5) so this give us:

  • x1 = -4
  • y1 = 10
  • x2 = -1
  • y2 = -5

This gives us:

\[ m = {(-5 - 10) \over (-1 - (-4))} = {-15 \over 3} = -5 \]

This means that:

\[ y = -5x + c \]

As we are only looking to find the gradient, we do not need to worry about finding the value of c.

The value of m is the coefficient of the x term which is -5.

So the gradient or slope of the line is -5.

Example 6) Find the gradient of the line shown below.

slope of a linear equation example 6 image

First we need to choose two points which lie exactly on the line.

We are going to choose (-6, 6) and (6, 0).

The equation to find the gradient is:

\[ m = {(y_2 - y_1) \over (x_2 - x_1)} \]

Our two points are (-6, 6) and (6, 0) so this give us:

  • x1 = -6
  • y1 = 6
  • x2 = 6
  • y2 = 0

This gives us:

\[ m = {(0 - 6) \over (6 - (-6))} = {-6 \over 12} = -{1 \over 2} \]

This means that:

\[ y = -{1 \over 2}x + c \]

As we are only looking to find the gradient, we do not need to worry about finding the value of c.

The value of m is the coefficient of the x term which is -½.

So the gradient or slope of the line is -½.

Slope of a Linear Equation Worksheets

We have created a series of worksheets to help you practice finding the slope or gradient of a linear equation.

Each sheet involves finding the slope from either an equation, two points or a line on a coordinate grid.

  • Sheet 1 involves finding a gradient from a linear equation in both slope-intercept and standard form.
  • Sheet 2 involves finding a slope from a set of two coordinates.
  • Sheets 3 and 4 involve finding a slope from a line on a coordinate grid
  • Sheet 5 involves a mixture of all of the above
gradient of a linear equation worksheet 1 gradient of a linear equation worksheet 2 gradient of a linear equation worksheet 3 gradient of a linear equation worksheet 4 gradient of a linear equation worksheet 5

Linear Equation Calculator

Our Linear Equation Calculator will find a linear equation from two points and display the equation both in slope-intercept form and standard form.

You can easily use the calculator to find the slope (m) of the linear equation.

However the best thing about our calculator is that it takes you through all the steps needed to find a linear equation from two points.

line equation from two points calculator image

More Recommended Math Worksheets

Take a look at some more of our worksheets similar to these.

Linear Equation with Two Points Support

If you need further support on solving linear equations, then try our dedicated support page.

On the page you will find:

  • clear definitions and explanations how to find a linear equation from two points
  • lots of worked examples
  • practice worksheets
  • link to our linear equation calculator
linear equation with two points header image

Linear Equations that are Parallel Support

If you are needing help with finding parallel linear equations or lines which are parallel to other lines then take a look at the link below.

On the page you will find:

  • how to quickly derive linear equations which are parallel
  • how to find parallel lines from linear equations which pass through specific coordinates
  • practice examples and worksheets
linear equations that are parallel header image

Negative Number Worksheets and Support

In order to find a linear equation from two points, you need to be proficient at adding and subtracting negative numbers.

Here are some of our resources which will help you to learn and practice these skills.

understanding negative numbers on a number line image negative number line worksheets image Adding positive and negative numbers image subtracting positive and negative numbers image

Factorising Quadratic Equation Worksheets

The sheets are arranged in order of difficulty to help introduce you into more complicated examples gradually.

Using these sheets will hopefully make factorising quadratic equations simple and straightforward.

The sheets cater for simpler and more complex quadratic equations.

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