Writing a Linear Equation from Two Points
Given two points, you can pin down the whole line: find the slope first, then use one point to get the intercept.
Reading the concept isn't enough. The score comes from practicing the real Bluebook-style format and finding the exact mistakes you keep making.
What the SAT tests here
The slope is $\frac{y_2 - y_1}{x_2 - x_1}$. Once you have it, substitute either point into $y = mx + b$ and solve for $b$.
How to solve one, step by step
Example: the line through $(1, 2)$ and $(3, 8)$.
- Slope: $\frac{8 - 2}{3 - 1} = 3$.
- Use $(1, 2)$: $2 = 3(1) + b \Rightarrow b = -1$.
- Equation: $y = 3x - 1$.
The mistakes that cost points
- Using the wrong point for b. After finding the slope, substitute an actual point — don't guess the intercept.
- Subtracting coordinates in the wrong order. Keep $x$'s and $y$'s in the same order top and bottom of the slope fraction.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
In the xy-plane, line $l$ is perpendicular to $y = -\frac{1}{3}x + 1$ and contains the point $(2, -4)$. What is the equation of $l$?
- A$y = 3x$
- B$y = 3x - 10$
- C$y = -3x + 2$
- D$y = \frac{1}{3}x - \frac{14}{3}$
Show solution
Answer: B, $y = 3x - 10$. The slope of the given line is $-\frac{1}{3}$, so the slope of any perpendicular line is its negative reciprocal: $3$. Substitute the point $(2, -4)$ into $y = 3x + b$: $-4 = 3(2) + b$, so $b = -10$. Equation: $y = 3x - 10$.
Medium
Line $l$ is perpendicular to $y = 9x - 3$ and passes through the point $(18, -8)$. What is the equation of $l$?
- A$y = \frac{1}{9}x - 10$
- B$y = -\frac{1}{9}x$
- C$y = -9x + 154$
- D$y = -\frac{1}{9}x - 6$
Show solution
Answer: D, $y = -\frac{1}{9}x - 6$. The slope of the given line is $9$, so the slope of any perpendicular line is its negative reciprocal: $-\frac{1}{9}$. Substitute the point $(18, -8)$ into $y = -\frac{1}{9}x + b$: $-8 = -\frac{1}{9}(18) + b$, which gives $-8 = -2 + b$, so $b = -6$. Equation: $y = -\frac{1}{9}x - 6$.
Hard
Line $l$ is perpendicular to $y = -\frac{3}{4}x - 1$ and passes through the point $(3, 2)$. What is the equation of $l$?
- A$y = \frac{4}{3}x$
- B$y = -\frac{4}{3}x + 6$
- C$y = \frac{3}{4}x - \frac{1}{4}$
- D$y = \frac{4}{3}x - 2$
Show solution
Answer: D, $y = \frac{4}{3}x - 2$. The slope of the given line is $-\frac{3}{4}$, so the slope of any perpendicular line is its negative reciprocal: $\frac{4}{3}$. Substitute the point $(3, 2)$ into $y = \frac{4}{3}x + b$ and solve for $b$. Equation: $y = \frac{4}{3}x - 2$.
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