SAT Math: Linear equations in two variables

Linear equations in two variables, such as $a x + b y = c$ .

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

Practise linear equations in two variables →Adaptive SAT module

What the test covers

Standard form ax + by = c; intercepts
Interpreting coefficients in context

Key ideas

Intercepts: set $x = 0$ for the $y$ -intercept and $y = 0$ for the $x$ -intercept.
Solve for $y$ to get slope-intercept form: $y = - \frac{a}{b} x + \frac{c}{b}$ .
In context, each coefficient is a price or rate and $c$ is the total.

Common mistakes

The slope of $3 x - 5 y = 30$ is $\frac{3}{5}$ , not 3.
Answer in the units the question asks for.

Do it in Desmos

Graph the equation directly (Desmos accepts 3x-5y=30) and click the intercepts.

Worked example

In the $x y$ -plane, the graph of $3 x - 4 y = - 12$ crosses the $y$ -axis at $(0, k)$ . What is the value of $k$ ?

$- 3$
$3$
$- 4$
$4$

Answer: B: $3$

On the $y$ -axis $x = 0$ , so $- 4 k = - 12$ and $k = 3$ .

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

Adult tickets to a concert cost $10 and child tickets cost $6. A group spent exactly $60 and bought 3 adult tickets. How many child tickets did they buy?

Show the answer and solution

Answer: C: $5$

$10 (3) + 6 c = 60$ , so $6 c = 30$ and $c = 5$ .

Question 2

Adult tickets to a concert cost $8 and child tickets cost $5. A group spent exactly $50 and bought 5 adult tickets. How many child tickets did they buy?

Show the answer and solution

Answer: B: $2$

$8 (5) + 5 c = 50$ , so $5 c = 10$ and $c = 2$ .

Question 3

Line $ℓ$ passes through $(2, 5)$ and is parallel to the line $4 x - 2 y = 7$ . What is the $y$ -coordinate of the $y$ -intercept of line $ℓ$ ?

Student-produced response: type your answer.

Show the answer and solution

Answer: 1

$4 x - 2 y = 7$ is $y = 2 x - \frac{7}{2}$ , slope 2. So $ℓ$ : $y - 5 = 2 (x - 2)$ , $y = 2 x + 1$ ; the $y$ -intercept is $(0, 1)$ .

Question 4

Adult tickets to a school play cost $6 and child tickets cost $4. A family spent exactly $48 and bought 4 adult tickets. How many child tickets did they buy?