Lesson 2: Writing Equations in 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

They are really the same formula, but they are used for different purposes:

• We use the slope formula to calculate the slope of a line when we know two points on the line. That is, we know $(x_1, y_1)$ and $(x_2, y_2)$, and we are looking for $m$. The slope formula is $m = \frac{y_2 - y_1}{x_2 - x_1}$.
• We use the point-slope formula to find the equation of a line. That is, we know $(x_1, y_1)$ and $m$, and we are looking for an equation like $y = mx + b$. The point-slope formula is $m = \frac{y - y_1}{x - x_1}$.

Examples

• To find the slope of a line passing through $(3, 6)$ and $(5, 12)$, use the slope formula: $m = \frac{12 - 6}{5 - 3} = \frac{6}{2} = 3$.
• A line has a slope of $5$ and passes through $(-1, 4)$. To find its equation, use the point-slope formula: $y - 4 = 5(x - (-1))$, which simplifies to $y = 5x + 9$.
• To find the equation of a line through $(2, 1)$ and $(4, 7)$, first find the slope $m = \frac{7-1}{4-2} = 3$. Then use the point-slope formula with $(2, 1)$: $y - 1 = 3(x - 2)$.

Explanation

The slope formula finds the value of the slope, $m$, using two known points. The point-slope formula uses that slope and one point to build the entire equation of the line. Think of it as finding the tool, then using the tool.

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 $y = \frac{1}{2}x - 5$.