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 · 4 with free solutions
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.
$a x + b = c x + d$ has exactly one solution when $a \neq c$ , no solution when $a = c$ and $b \neq d$ , 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: $3 (x - 4) = 3 x - 12$ .
'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

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

$7$
$5$
$- 2$
$- 5$

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.

Practice questions

Try each question before opening the solution. Desmos or your own calculator is allowed on every SAT Math question. Solutions to the first 3 are open to everyone; the rest need only a free account. Every answer here was checked independently before it was published.

Question 1

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: $3 x - 6 + 4 = x$ , so $2 x = 2$ and $x = 1$ .

Question 2

What value of $x$ satisfies $\frac{2 x + 5}{3} = \frac{x - 1}{2} + 4$ ?

Student-produced response: type your answer.

Show the answer and solution

Answer: 11

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

Question 3

What value of $x$ satisfies $\frac{2 x + 1}{6} = \frac{x}{4} + 2$ ?

Student-produced response: type your answer.

Show the answer and solution

Answer: 22

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