SAT Math: Linear equations in two variables

Linear equations in two variables, such as $ax+by=c$.

Algebra · Linear equations in two variables
Both modules
13 practice questions
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: $y=-\tfrac abx+\tfrac cb$.
In context, each coefficient is a price or rate and $c$ is the total.

Common mistakes

The slope of $3x-5y=30$ is $\tfrac35$, 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

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?

$5$
$6$
$4$
$8$

Answer: B: $6$

$6(4)+4c=48$, so $4c=24$ and $c=6$.

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

In the $xy$-plane, the graph of $3x-5y=30$ crosses the $y$-axis at the point $(0, b)$. What is the value of $b$?

$6$
$-6$
$-10$
$10$

Show the answer and solution

Answer: B: $-6$

At the $y$-intercept $x=0$: $-5y=30$, $y=-6$.

Question 2

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

Student-produced response: type your answer.

Show the answer and solution

Answer: 1

$4x-2y=7$ is $y=2x-\tfrac72$, slope 2. So $\ell$: $y-5=2(x-2)$, $y=2x+1$; the $y$-intercept is $(0,1)$.

Question 3

In the $xy$-plane, the graph of $6x-2y=12$ crosses the $y$-axis at $(0,k)$. What is the value of $k$?

$6$
$-2$
$2$
$-6$

Show the answer and solution

Answer: D: $-6$

On the $y$-axis $x=0$, so $-2k=12$ and $k=-6$.

Question 4

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

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

Show the answer and solution

Answer: B: $3$

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

Question 5

In the $xy$-plane, the graph of $6x+3y=-18$ crosses the $y$-axis at $(0,k)$. What is the value of $k$?

$-6$
$-3$
$6$
$3$

Show the answer and solution

Answer: A: $-6$

On the $y$-axis $x=0$, so $3k=-18$ and $k=-6$.

Question 6

In the $xy$-plane, the graph of $3x-5y=-15$ crosses the $y$-axis at $(0,k)$. What is the value of $k$?

$-3$
$-5$
$5$
$3$

Show the answer and solution

Answer: D: $3$

On the $y$-axis $x=0$, so $-5k=-15$ and $k=3$.

Question 7

In the $xy$-plane, the graph of $5x+3y=-15$ crosses the $y$-axis at $(0,k)$. What is the value of $k$?

$3$
$-3$
$5$
$-5$

Show the answer and solution

Answer: D: $-5$

On the $y$-axis $x=0$, so $3k=-15$ and $k=-5$.

Question 8

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?

$7$
$6$
$5$
$3$

Show the answer and solution

Answer: C: $5$

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

Question 9

Adult tickets to a concert cost $10 and child tickets cost $6. A group spent exactly $38 and bought 2 adult tickets. How many child tickets did they buy?

$2$
$5$
$3$
$4$

Show the answer and solution

Answer: C: $3$

$10(2)+6c=38$, so $6c=18$ and $c=3$.

Question 10

Adult tickets to a concert cost $10 and child tickets cost $6. A group spent exactly $70 and bought 4 adult tickets. How many child tickets did they buy?

$4$
$5$
$6$
$7$

Show the answer and solution

Answer: B: $5$

$10(4)+6c=70$, so $6c=30$ and $c=5$.

Question 11

Adult tickets to a concert cost $6 and child tickets cost $4. A group spent exactly $32 and bought 2 adult tickets. How many child tickets did they buy?

$2$
$4$
$6$
$5$

Show the answer and solution

Answer: D: $5$

$6(2)+4c=32$, so $4c=20$ and $c=5$.

Question 12

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?

$5$
$2$
$3$
$4$

Show the answer and solution

Answer: B: $2$

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

Keep going

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Next topic: Systems of linear equations
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