Find the Equation of a Line From Two Points

Each linear equations worksheet on this page shows four graphs on a coordinate plane, each with two points labeled, and students find the equation in slope-intercept form by calculating both the slope and y-intercept.

What is Slope Intercept Form?

Equations that describe a line (i.e., linear equations) are often shown in slope intercept form...

Equations in point slope are a great way to describe a line if you know the slope and at least one point on the line. You start with the point slope equation and substitute the slope constant for m, and then the x and y coordinates for the point for x₁ and y₁.

If you are graphing linear equations, the worksheets on this page provide great practice resources for middle school students. You can also use a blank coordinate plane to graph your own equations, or try working with the slope calculator to see how to find the slope from two points.

How to Find Linear Equations in Point Slope Form From Two Points

If you have two points that define a line, you can figure the find an equation in slope intercept form by following these steps :

1. Find the slope of the line using the rise-over-run slope equation (See below).
2. Substitute the m value in the y=mx+b slope intercept equation with the calculated slope.
3. Take the x and y values from one of the starting points, and substitute them in that equation, then solve for b. This solved value is the y-intercept.
4. Go back to a simple y=mx+b form and substitute the calculated slope for m and the calculated y-intercept for b. The result is a slope intercept equation for the line passing through the original two points.

You may also calculate the equation for a line by changing the slope independently (either as a slope fraction or a slope decimal), or by entering a new y intercept. If a new slope is entered, the slope calculator will move one of the points so that the equation matches the new line. If a new y intercept is entered, the slope will remain the same but the calculator will move the two points to shift the line to match the new y intercept.

What is the Slope of a Line?

The slope of a line is a mathematical measurement of how steep a line drawn on a graph appears, and this value is usually shown as the variable m in an equation in slope intercept form, y=mx+b.

Slope is defined as the ratio of vertical (y-axis) change over a given amount of horizontal (x-axis) change, often remembered more simply as a fraction describing rise over run or the rate of change. Slope is usually shown as an fraction, often an improper fraction, but it can also be represented as a mixed fraction or decimal number in some situations.

If a line is sloping up and to the right, it is rising as you look left-to-right across the x-axis. The rise in this case is positive, and such a line will have a positive slope.

If a line is sloping down and to the right, it is falling as you look left-to-right across the x-axis. The rise in this case is negative (the line is "falling"), and such a line will have a negative slope.

How to Find the Slope of a Line from Two Points

Given two points that define a line on a Cartesian coordinate plane, the slope of the line is calculated using the slope equation below:

By starting with two points (x₁,y₁) and (x₂,y₂), the substitute the values into the equation to calculate the "rise" on the top and the "run" on the bottom. It doesn't matter which point is used as (x₁,y₁) or (x₂,y₂), but it is super important that you consistently use the coordinates from each point once you choose. For example, if you select one point such as (5, 6), be sure to use 6 as the minuend of the subtraction on the top of the equation, and 5 as the minuend of the subtraction on the bottom of the equation. When in doubt, use the slope calculator to check your work.

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If you are graphing linear equations, the worksheets on this page provide great practice resources for middle school students. You can also use a blank coordinate plane to graph your own equations, or try working with the slope calculator to see how different points, slope and y-intercept values can be combined to make an equation in slope intercept form.