Writing the Equation of a Line Using Two Points

Point-slope form of an equation is also useful if we know any two points through which a line passes. Suppose, for example, we know that a line passes through the points [latex]\left(0,\text{ }1\right)[/latex] and [latex]\left(3,\text{ }2\right)[/latex]. We can use the coordinates of the two points to find the slope.

[latex]\begin{array}{l}{m}=\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}}\\ \text{}{m}=\frac{2 - 1}{3 - 0}\hfill \\ \text{}{m}=\frac{1}{3}\hfill \end{array}[/latex]

Now we can use the slope we found and the coordinates of one of the points to find the equation for the line. Let’s use [latex](0, 1)[/latex] for our point.

[latex]\begin{array}{l}y-{y}_{1}=m\left(x-{x}_{1}\right)\\ y - 1=\frac{1}{3}\left(x - 0\right)\end{array}[/latex]

As before, we can use algebra to rewrite the equation in slope-intercept form.

[latex]\begin{array}{lll}y - 1=\frac{1}{3}\left(x - 0\right)\hfill & \hfill \\ y - 1=\frac{1}{3}x\hfill & \text{Distribute the }\frac{1}{3}.\hfill \\ \text{}y=\frac{1}{3}x+1\hfill & \text{Add 1 to each side}.\hfill \end{array}[/latex]

Both equations describe the line graphed below.

Writing the Equation of a Line Using a Graph

So far we have written point-slope form and slope-intercept form of a line when given a point and the slope as well as when given two points. Sometimes the only information we are provided is the graph of the line. Let’s look into how we can write the point-slope form and slope-intercept form of a line when only given a graph.

Look at the graph of the function [latex]f[/latex] given below.

We are not given the slope of the line, but we can choose any two points on the line to find the slope. Let’s choose [latex](0,7)[/latex] and [latex](4,4)[/latex].

Now we can substitute the slope and the coordinates of one of the points into the point-slope form.

If we want to rewrite the equation in the slope-intercept form, we would find

If we want to find the slope-intercept form without first writing the point-slope form, we could have recognized that the line crosses the y-axis when the output value is [latex]7[/latex]. Therefore, [latex]b=7[/latex]. We now have the initial value [latex]b[/latex] and the slope [latex]m[/latex] so we can substitute [latex]m[/latex] and [latex]b[/latex] into the slope-intercept form of a line.

So the function is [latex]f(x)=-\frac{3}{4}x+7[/latex], and the linear equation would be [latex]y=-\frac{3}{4}x+7[/latex].

Write an equation for a linear function given a graph of [latex]f[/latex] shown below.

Show Solution

Identify two points on the line, such as [latex](0, 2)[/latex] and [latex](-2, -4)[/latex]. Use the points to calculate the slope.

[latex]\begin{array}{ccl} m & = & \frac{y_2-y_1}{x_2-x_1} \\ & = & \frac{-4-2}{-2-0} \\ & = & \frac{-6}{-2} \\ & = & 3 \end{array}[/latex]

Substitute the slope and the coordinates of one of the points into the point-slope form.

[latex]\begin{array}{ccl} y - y_1 & = & m(x - x_1) \\ y - (-4) & = & 3(x - (-2)) \\ y + 4 & = & 3(x + 2) \end{array}[/latex]

We can use algebra to rewrite the equation in the slope-intercept form.

[latex]\begin{array}{ccl} y + 4 & = & 3(x + 2) \\ y + 4 & = & 3x + 6 \\ y & = & 3x + 2 \end{array}[/latex]

Writing and Interpreting an Equation for a Linear Function

Now we can choose which method to use to write equations for linear functions based on the information we are given. That information may be provided in the form of a graph, a point and a slope, two points, and so on. Let’s try a few more examples when we are given the details of a linear function in different ways.