SAT Math: Linear equations in one variable

Linear equations in one variable: the most common Algebra question on the digital SAT.

Algebra · Linear equations in one variable
Both modules
15 practice questions
Calculator allowed (Desmos)

Practise linear equations in one variable →Adaptive SAT module

What the test covers

Solve, including equations with fractions and a parameter
No solution / infinitely many solutions conditions
Build an equation from a context

Key ideas

Undo operations in reverse order; clear fractions by multiplying every term by the common denominator.
\(ax+b=cx+d\) has exactly one solution when \(a\ne c\), no solution when \(a=c\) and \(b\ne d\), and infinitely many when \(a=c\) and \(b=d\).
Word problems: name the unknown, write one equation, solve, then answer the question actually asked.

Common mistakes

Distribute to every term in the bracket: \(3(x-4)=3x-12\).
'No solution' questions ask about coefficients, not about solving.

Do it in Desmos

Type both sides as y1 = left side and y2 = right side; the x-coordinate of the intersection is the solution.

Worked example

If \(3(x-4)+2=5x-16\), what is the value of \(x\)?

\(3\)
\(7\)
\(-3\)
\(\frac{9}{4}\)

Answer: A: \(3\)

\(3x-12+2=5x-16\), so \(3x-10=5x-16\), \(6=2x\), \(x=3\).

Practice questions

Try each question before opening the solution. Desmos or your own calculator is allowed on every SAT Math question. Every answer here was re-checked independently by computer before it was published.

Question 1

What value of \(x\) satisfies \(\dfrac{2x+5}{3}=\dfrac{x-1}{2}+4\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: 11

Multiply by 6: \(4x+10=3x-3+24\), so \(x=11\).

Question 2

In the equation \(4(2x-k)=8x-12\), \(k\) is a constant. For what value of \(k\) does the equation have infinitely many solutions?

\(3\)
\(12\)
\(-3\)
\(4\)

Show the answer and solution

Answer: A: \(3\)

\(8x-4k=8x-12\) holds for every \(x\) exactly when \(4k=12\), i.e. \(k=3\).

Question 3

In the equation \(5x+7=ax-2\), \(a\) is a constant. For what value of \(a\) does the equation have no solution?

\(7\)
\(5\)
\(-2\)
\(-5\)

Show the answer and solution

Answer: B: \(5\)

\((5-a)x=-9\). If \(a=5\) this says \(0=-9\): no solution. For any other \(a\) there is exactly one solution.

Question 4

If \(0.4(x+15)=0.25x+9\), what is the value of \(x\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: 20

\(0.4x+6=0.25x+9\), so \(0.15x=3\) and \(x=20\).

Question 5

If \(2(x-2)+6=x\), what is the value of \(x\)?

\(2\)
\(-2\)
\(0\)
\(10\)

Show the answer and solution

Answer: B: \(-2\)

Expand: \(2x-4+6=x\), so \(x=-2\) and \(x=-2\).

Question 6

If \(3(x-4)+2=x\), what is the value of \(x\)?

\(-5\)
\(7\)
\(5\)
\(8\)

Show the answer and solution

Answer: C: \(5\)

Expand: \(3x-12+2=x\), so \(2x=10\) and \(x=5\).

Question 7

If \(2(x-3)-4=7x\), what is the value of \(x\)?

\(-2\)
\(0\)
\(2\)
\(-4\)

Show the answer and solution

Answer: A: \(-2\)

Expand: \(2x-6-4=7x\), so \(-5x=10\) and \(x=-2\).

Question 8

If \(2(x-1)-5=x\), what is the value of \(x\)?

\(-7\)
\(-3\)
\(7\)
\(9\)

Show the answer and solution

Answer: C: \(7\)

Expand: \(2x-2-5=x\), so \(x=7\) and \(x=7\).

Question 9

If \(3(x-2)+4=x\), what is the value of \(x\)?

\(5\)
\(1\)
\(-1\)
\(4\)

Show the answer and solution

Answer: B: \(1\)

Expand: \(3x-6+4=x\), so \(2x=2\) and \(x=1\).

Question 10

What value of \(x\) satisfies \(\dfrac{4x+3}{6}=\dfrac{x}{2}+5\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: 27

Multiply by \(12\): \(2(4x+3)=6x+60\), so \(2x=54\) and \(x=27\).

Question 11

What value of \(x\) satisfies \(\dfrac{2x+2}{5}=\dfrac{x}{2}+1\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: -6

Multiply by \(10\): \(2(2x+2)=5x+10\), so \(-1x=6\) and \(x=-6\).

Question 12

What value of \(x\) satisfies \(\dfrac{2x+1}{6}=\dfrac{x}{4}+2\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: 22

Multiply by \(24\): \(4(2x+1)=6x+48\), so \(2x=44\) and \(x=22\).

Question 13

What value of \(x\) satisfies \(\dfrac{2x+2}{6}=\dfrac{x}{4}+3\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: 32

Multiply by \(24\): \(4(2x+2)=6x+72\), so \(2x=64\) and \(x=32\).

Question 14

What value of \(x\) satisfies \(\dfrac{2x+3}{6}=\dfrac{x}{4}+5\)?

Student-produced response: type your answer.

Show the answer and solution

Answer: 54

Multiply by \(24\): \(4(2x+3)=6x+120\), so \(2x=108\) and \(x=54\).

Keep going

Previous topic: Circles
Next topic: Linear functions
All SAT Math skills · Adaptive SAT Math module simulator · Official SAT practice (Bluebook, Khan Academy)

SAT is a trademark registered by College Board, which is not affiliated with, and does not endorse, IB Math Revision. Every question here is our own.