Solving Systems of Equations on the SAT
A system gives you two equations and asks for the values that satisfy both at once. Two reliable methods cover almost every SAT question: substitution and elimination.
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
Elimination adds or subtracts the equations to cancel a variable; substitution solves one equation for a variable and plugs it into the other. Either works — pick whichever makes the numbers cleaner, and remember to find both variables.
How to solve one, step by step
Example: solve $2x + y = 7$ and $x - y = 2$.
- Add the equations so $y$ cancels: $3x = 9 \Rightarrow x = 3$.
- Substitute back: $3 - y = 2 \Rightarrow y = 1$.
- Solution: $(3, 1)$.
The mistakes that cost points
- Adding without aligning like terms. Line up the $x$'s, $y$'s, and constants before combining equations.
- Substituting into the wrong place. Plug your expression into the other equation, not the one you solved.
- Stopping at one variable. The SAT often asks for $x + y$ or $y$ — find both before answering.
Practice questions
Try these the way you would on test day, then open the solution to check your method.
Easy
A linear function passes through the points $(1, 6)$ and $(3, 14)$. What is the equation of the line in slope-intercept form?
- A$y = 4x + 2$
- B$y = 8x - 2$
- C$y = \frac{1}{4}x + \frac{23}{4}$
- D$y = 4x + 6$
Show solution
Answer: A, $y = 4x + 2$. Use the slope formula with the two points $(1, 6)$ and $(3, 14)$: slope $= \frac{14 - (6)}{3 - (1)} = 4$. Use slope-intercept form $y = mx + b$ and substitute one of the points to solve for $b$: $6 = 4(1) + b$, so $b = 2$. Equation: $y = 4x + 2$.
Medium
If for a linear function $f$, $f(-5) = 0$ and $f(4) = -9$, what is the value of $f(0)$?
- A0
- B-1
- C5
- D-5
Show solution
Answer: D, -5. From $f(-5) = 0$ and $f(4) = -9$ we get points $(-5, 0)$ and $(4, -9)$. Slope: $m = \frac{-9 - (0)}{4 - (-5)} = \frac{-9}{9} = -1$. Y-intercept: $b = 0 - (-1)(-5) = -5$. So $f(x) = -1x + (-5)$, and $f(0) = (-1)(0) + (-5) = -5$.
Hard
A linear function passes through the points $(-2, 3)$ and $(6, 9)$. What is the equation of the line in slope-intercept form?
- A$y = \frac{3}{4}x + \frac{9}{2}$
- B$y = \frac{3}{4}x + 3$
- C$y = \frac{3}{4}x + 9$
- D$y = \frac{4}{3}x + \frac{17}{3}$
Show solution
Answer: A, $y = \frac{3}{4}x + \frac{9}{2}$. Use the slope formula with the two points $(-2, 3)$ and $(6, 9)$: slope $= \frac{9 - (3)}{6 - (-2)} = \frac{3}{4}$. Use slope-intercept form $y = mx + b$ and substitute one of the points to solve for $b$: $3 = \frac{3}{4}(-2) + b$, so $b = \frac{9}{2}$. Equation: $y = \frac{3}{4}x + \frac{9}{2}$.
Find your exact gaps
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