Lesson 2: Point-Slope Form

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 convert from point-slope form to slope-intercept form, solve for $y$:

Examples

Property

To find an equation for the line of slope $m$ passing through the point $(x_1, y_1)$, we use the point-slope formula

or

To Fit a Line through Two Points.

1. Compute the slope between the two points.
2. Substitute the slope and either point into the point-slope formula.

Examples

• To find the equation for a line with slope $m=4$ passing through $(1,3)$, use the formula $y-y_1 = m(x-x_1)$ to get $y-3 = 4(x-1)$.
• To find the equation for the line passing through $(2,5)$ and $(4,11)$, first calculate the slope: $m = \frac{11-5}{4-2} = \frac{6}{2} = 3$. Then use the point-slope formula with the point $(2,5)$: $y-5 = 3(x-2)$.
• A line passes through $(-2, 8)$ with a slope of $-3$. To write its equation in slope-intercept form, start with point-slope: $y-8 = -3(x - (-2))$. This becomes $y-8 = -3(x+2)$. Distribute to get $y-8 = -3x-6$, then add 8 to both sides: $y = -3x+2$.

Explanation

This formula is the most direct way to write a line's equation when you have its slope and any point it passes through. It's built directly from the definition of slope, $m = \frac{\text{rise}}{\text{run}}$.