Lesson 52: Determining the Equation of a Line Given Two Points

New Concept

The form $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a point on the line, is called the point-slope form.

What’s next

Next, you’ll use this form to write equations for lines when given any two points, applying it to solve problems.

Property

The form $y - y_1 = m(x - x_1)$, where $m$ is the slope and $(x_1, y_1)$ is a point on the line is called the point-slope form of a line.

Examples

Write the equation for a line with slope 3 passing through (2, 4): $y - 4 = 3(x - 2)$.
Write the equation for a line with slope -5 passing through (1, 8): $y - 8 = -5(x - 1)$.
Write the equation for a line with slope $\frac{1}{2}$ passing through (-3, 6): $y - 6 = \frac{1}{2}(x + 3)$.

Explanation

Think of this as your instant recipe for a line! If you have one point and the slope (the 'steepness'), you can plug them straight into this formula. It is the fastest way to write the equation without finding the y-intercept first. Super handy for quick equations!

Property

First, find the slope with $m = \frac{y_2 - y_1}{x_2 - x_1}$. Then, use the point-slope form $y - y_1 = m(x - x_1)$ to write the equation.

Examples

Find the equation for a line through (2, 3) and (5, 9): First, $m = \frac{9 - 3}{5 - 2} = \frac{6}{3} = 2$. Then, use a point: $y - 9 = 2(x - 5)$.
Find the equation for a line through (1, -3) and (4, 5): First, $m = \frac{5 - (-3)}{4 - 1} = \frac{8}{3}$. Then, use a point: $y - 5 = \frac{8}{3}(x - 4)$.

Explanation

Got two points but no slope? No problem! First, calculate the 'rise over run' between the two points to find your slope. Then, just pick one of those points, grab your new slope, and pop them right into the point-slope formula. You've officially built a line from scratch!

Property

To graph a line with a known slope and point, first plot the given point. Then, use the slope (rise over run) from that point to find a second point and draw a line through both.

Examples

Graph a line with slope 4 and point (3, 5): Plot (3, 5). From there, go up 4 and right 1 to find point (4, 9). Draw the line.
Graph a line with slope $-\frac{2}{3}$ and point (2, 4): Plot (2, 4). From there, go down 2 and right 3 to find point (5, 2). Draw the line.

Explanation

It is like a treasure map! Start at your given point ('X' marks the spot). The slope tells you where to go next: 'rise' is your up/down count, and 'run' is your right count. Plot your second point where you land and connect the dots to reveal the treasure path!