Writing a Linear Equation from a Point and Slope
When you know one point and the slope, point-slope form gets you the equation in a single step.
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
Point-slope form is $y - y_1 = m(x - x_1)$. Plug in the slope and the point, then simplify to $y = mx + b$ if the question wants that form.
How to solve one, step by step
Example: slope $4$, through $(2, 5)$.
- Point-slope: $y - 5 = 4(x - 2)$.
- Distribute and simplify: $y = 4x - 3$.
The mistakes that cost points
- Forgetting to subtract x1. It's $m(x - x_1)$ — the point's $x$ gets subtracted inside the parentheses.
- Swapping x and y. Match each coordinate to the right variable when substituting.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
The line $6x + 3y = 12$ has the same slope as the line $y = kx - 1$. What is the value of $k$?
- A$4$
- B$-2$
- C$-6$
- D$2$
Show solution
Answer: B, $-2$. Convert $6x + 3y = 12$ to slope-intercept form. Subtract $6x$: $3y = -6x + 12$. Divide by $3$: $y = -2x + 4$. The slope is $-2$, so $k = -2$.
Medium
A line is given by the equation $3x + ky = 12$. When this equation is written in slope-intercept form, the slope is $-\frac{3}{4}$. What is the value of $k$?
- A$3$
- B$4$
- C$\frac{1}{4}$
- D$-4$
Show solution
Answer: B, $4$. Rewrite $3x + ky = 12$: subtract $3x$ to get $ky = -3x + 12$, then divide by $k$: $y = -\frac{3}{k}x + \frac{12}{k}$. Set $-\frac{3}{k} = -\frac{3}{4}$, so $k = 4$.
Hard
Two lines are given: Line 1 is $6x - 4y = k$ and Line 2 is $y = \frac{3}{2}x - 5$. If the two lines are the same, what is the value of $k$?
- A$5$
- B$20$
- C$-20$
- D$-5$
Show solution
Answer: B, $20$. Convert Line 1 to slope-intercept form: $-4y = -6x + k$, so $y = \frac{6}{4}x - \frac{k}{4} = \frac{3}{2}x - \frac{k}{4}$. For the lines to be identical, $-\frac{k}{4} = -5$, so $k = 20$.
Find your exact gaps
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