Solving Linear Equations on the SAT

SAT Math · Algebra · Updated June 2026

Single-variable linear equations are one of the most frequently tested skills on the SAT Math section — they show up on nearly every test, and they're the foundation for almost everything else in algebra.

Reading how to solve them 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

A linear equation has the variable to the first power only — no $x^2$, no square roots, no variables in a denominator. Your job is to isolate the variable. On the SAT, the equation is usually dressed up with distribution (parentheses) and fractions so that you have to do a couple of clean steps before the answer appears.

How to solve one, step by step

Example: solve $3(x - 4) = 18$.

Distribute the 3 across the parentheses: $3x - 12 = 18$.
Move the constant by adding 12 to both sides: $3x = 30$.
Divide both sides by 3: $x = 10$.

The mistakes that cost points

Not distributing first. Treating $3(x - 4)$ as if it were $3x - 4$ throws off every step that follows.
Moving a term without changing its sign. When $-12$ moves to the other side it becomes $+12$. Dropping that sign flip is the most common slip.
Multiplying when you should divide. To undo $3x$ you divide by 3 — multiplying instead lands on a tempting wrong answer.

Practice questions

Try these the way you would on test day, then open the solution to check your method.

Easy

The equation $5(x + k) - 5 = 20$ has the solution $x = 3$. What is the value of $k$?

A$k = 10$
B$k = 2$
C$k = 6$
D$k = 50$

Show solution

Answer: B, $k = 2$. Substitute $x = 3$: $5(3 + k) - 5 = 20$. Distribute: $15 + 5k - 5 = 20$. Simplify: $5k + 10 = 20$. Subtract 10: $5k = 10$. Divide by 5: $k = 2$.

Medium

The equation $2(3x + k) = 22$ has the solution $x = 2$. What is the value of $k$?

A$k = 17$
B$k = 20$
C$k = 5$
D$k = 10$

Show solution

Answer: C, $k = 5$. Substitute $x = 2$: $2(3(2) + k) = 22 \Rightarrow 2(6 + k) = 22$. Distribute: $12 + 2k = 22$. Subtract 12: $2k = 10$. Divide by 2: $k = 5$.

Hard

Solve for $x$: $3x + 22 = kx + 4$

A$x = kx - 6$
B$x = \frac{kx - 18}{3}$
C$x = \frac{18}{k - 3}$
DThe correct solution is not given

Show solution

Answer: C, $x = \frac{18}{k - 3}$. Start with $3x + 22 = kx + 4$. Subtract $kx$ and subtract $22$ from both sides: $(3 - k)x = 4 - 22$, so $(3 - k)x = -18$. Divide both sides by $(3 - k)$: $x = \frac{-18}{3 - k}$. Multiply numerator and denominator by $-1$ to get the standard form: $x = \frac{18}{k - 3}$.

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