Lesson 4.6: Find the Equation of a Line

New Concept

In this lesson, you'll learn to write the equation of a line using different starting points. We'll master two key forms: the slope-intercept form, $y = mx + b$, and the point-slope form, $y - y_1 = m(x - x_1)$.

What’s next

Soon, we’ll work through examples for each method. You will then apply these skills in a series of practice cards and challenge problems.

Property

We can easily determine the slope and intercept of a line if the equation was written in slope-intercept form, $y = mx + b$.
Now, we will do the reverse—we will start with the slope and y-intercept and use them to find the equation of the line.
To find an equation of a line with a given slope and y-intercept, substitute the slope ($m$) and the y-coordinate of the y-intercept ($b$) into the slope-intercept form, $y = mx + b$.

Examples

• Find the equation of a line with slope 4 and y-intercept $(0, 2)$. We substitute $m=4$ and $b=2$ into $y = mx + b$ to get the equation $y = 4x + 2$.

• Find the equation of a line with slope $-5$ and y-intercept $(0, -1)$. We substitute $m=-5$ and $b=-1$ into $y = mx + b$ to get the equation $y = -5x - 1$.

Property

The point-slope form of an equation of a line with slope $m$ and containing the point $(x_1, y_1)$ is

We can use the point-slope form of an equation to find an equation of a line when we are given the slope and one point.

Examples

• A line has a slope of 5 and passes through the point $(2, 8)$. Its equation in point-slope form is $y - 8 = 5(x - 2)$.

• A line has a slope of $-\frac{3}{4}$ and passes through the point $(-4, 1)$. Its equation is $y - 1 = -\frac{3}{4}(x - (-4))$, which simplifies to $y - 1 = -\frac{3}{4}(x + 4)$.

Property

To find an equation of a line with a given slope and a point, substitute the slope and the coordinates of the point into the point-slope form, $y - y_1 = m(x - x_1)$. Then, rewrite the equation in slope-intercept form, $y = mx + b$.

Examples

• Find the equation of a line with slope 3 that contains the point $(2, 5)$. Start with $y - 5 = 3(x - 2)$. Distribute to get $y - 5 = 3x - 6$. The final equation is $y = 3x - 1$.

• Find the equation of a line with slope $m = \frac{1}{2}$ that passes through $(6, -2)$. Start with $y - (-2) = \frac{1}{2}(x - 6)$. Simplify to $y + 2 = \frac{1}{2}x - 3$. The final equation is $y = \frac{1}{2}x - 5$.

Property

When given two points, first find the slope using the formula $m = \frac{y_2 - y_1}{x_2 - x_1}$. Then, choose one of the points and use the point-slope form, $y - y_1 = m(x - x_1)$, to find the equation. Finally, write the equation in slope-intercept form.

Examples

• Find the equation of a line passing through $(2, 3)$ and $(4, 7)$. First, find the slope: $m = \frac{7-3}{4-2} = \frac{4}{2} = 2$. Using the point $(2, 3)$, we have $y - 3 = 2(x - 2)$. The final equation is $y = 2x - 1$.

• Find the equation of a line passing through $(-1, 5)$ and $(3, -3)$. First, find the slope: $m = \frac{-3-5}{3-(-1)} = \frac{-8}{4} = -2$. Using the point $(-1, 5)$, we have $y - 5 = -2(x - (-1))$. The final equation is $y = -2x + 3$.