SAT Math: Systems of linear equations

Systems of two linear equations.

Algebra · Systems of two linear equations in two variables
Both modules
13 practice questions · 4 with free solutions
Calculator allowed (Desmos)

Practise systems of linear equations →Adaptive SAT module

What the test covers

Elimination and substitution; Desmos intersection
No / one / infinitely many solutions from coefficient ratios

Key ideas

Solve by elimination (add or subtract multiples) or substitution.
One solution: different slopes. No solution: same slope, different intercepts (parallel). Infinitely many: the same line.
For 'no solution' with a constant $k$ , make the $x$ and $y$ coefficients proportional and the constants not.

Common mistakes

A question asking for $x + y$ may be answerable by adding the equations directly.
Proportional coefficients alone are not enough for infinitely many solutions: the constants must match too.

Do it in Desmos

Graph both lines; the intersection is the solution. Parallel lines mean no solution.

Worked example

If $2 x + 5 y = 25$ and $x - 2 y = - 1$ , what is the value of $x + y$ ?

Answer: A: $8$

Solving the system (by elimination or with Desmos) gives $x = 5$ and $y = 3$ , so $x + y = 8$ .

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 $2 x + 5 y = 37$ and $2 x + y = 17$ , what is the value of $x + y$ ?

$6$
$5$
$11$
$1$

Show the answer and solution

Answer: C: $11$

Solving the system (by elimination or with Desmos) gives $x = 6$ and $y = 5$ , so $x + y = 11$ .

Question 2

In the system $5 x + k y = 2$ and $10 x - 4 y = 5$ , $k$ is a constant. The system has no solution. What is the value of $k$ ?

Student-produced response: type your answer.

Show the answer and solution

Answer: -2

No solution means parallel, distinct lines: the $x$ - and $y$ -coefficients are in the same ratio ( $5 : 10 = k : - 4$ , so $k = - 2$ ) but the constants are not ( $2 \times 2 \neq 5$ ).

Question 3

The system $a x + 2 y = 8$ and $3 x + b y = 12$ , where $a$ and $b$ are constants, has infinitely many solutions. What is the value of $a + b$ ?

Student-produced response: type your answer.

Show the answer and solution

Answer: 5

The two equations must be multiples of each other: $\frac{a}{3} = \frac{2}{b} = \frac{8}{12} = \frac{2}{3}$ . So $a = 2$ , $b = 3$ , $a + b = 5$ .