SAT Linear Equations in Two Variables Worksheet

For a line in standard form, set $x = 0$ to find the $y$ -intercept and set $y = 0$ to find the $x$ -intercept. To read the slope, rearrange the equation into $y = m x + b$ .

Written by Luke Finigan

SAT Linear Equations in Two Variables Worksheet, page 1: standard form, intercepts, slope, and equations from word problems

These 12 questions cover intercepts, standard form, slope, perpendicular lines, and equations from word problems. Use the printable student copy or the PDF with explained answers.

12 questions · about 25 minutes · Algebra › Linear equations in two variables

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Translate between a line’s equation and its points

In $A x + B y = C$ , a point is a solution only if substituting both coordinates makes the equation true. Keep the order $(x, y)$ and the signs. A graph’s intercepts are points, not just the constants in the equation.

When $B \neq = 0$ , solving for $y$ gives $y = - \frac{A}{B} x + \frac{C}{B}$ . The slope is $- \frac{A}{B}$ , so the coefficient of $x$ in standard form cannot be read directly as the slope. If $B = 0$ and $A \neq = 0$ , the line is vertical and its slope is undefined.

For a cost equation, attach each coefficient to the correct item and include its units. Test a purchase that should fit the budget. For perpendicular nonvertical lines, the slopes multiply to $- 1$ ; horizontal and vertical lines are the separate perpendicular case.

Try it yourself

What is the slope of

3 x + 2 y = 12

3

- \frac{3}{2}

\frac{2}{3}

Choose an answer.

Questions

Question 1
What is the $y$ -intercept of the graph of $4 x + 5 y = 40$ in the $x y$ -plane?
C. The $y$ -intercept is where $x = 0$ . Then $5 y = 40$ , so $y = 8$ . The point $(10, 0)$ is the $x$ -intercept.
Question 2
Which point lies on the graph of $2 x - 3 y = 12$ in the $x y$ -plane?
D. Substitute each point. For $(6, 0)$ , $2 (6) - 3 (0) = 12$ . The point $(0, 4)$ gives $- 12$ because the $y$ -term is subtracted; the $y$ -intercept is $(0, - 4)$ .
Question 3
A store sells notebooks for $3 each and folders for $2 each. Maya spends exactly $24 on $n$ notebooks and $f$ folders. Which equation represents this situation?
D. The notebooks cost $3 n$ dollars and the folders cost $2 f$ dollars. Together they cost 24 dollars, so $3 n + 2 f = 24$ . Choice A attaches each price to the wrong item.
Question 4
Jaya earned $600 last month from $t$ hours of tutoring at $15 per hour and $b$ hours at a bookstore at $10 per hour, so $15 t + 10 b = 600$ . If she tutored for 20 hours, how many hours did she work at the bookstore?
B. Substitute $t = 20$ : $300 + 10 b = 600$ , so $10 b = 300$ and $b = 30$ . Choice D, 60, is the number of bookstore hours if she had not tutored at all.
Question 5
What is the slope of the graph of $6 x + 2 y = 7$ in the $x y$ -plane?
A. Solve for $y$ : $2 y = - 6 x + 7$ , so $y = - 3 x + \frac{7}{2}$ . The slope is the coefficient of $x$ , which is −3.
Question 6
Which equation represents the line shown in the $x y$ -plane?
A. The line passes through (0, 6) and (4, 0). Both points satisfy $3 x + 2 y = 12$ : $3 (0) + 2 (6) = 12$ and $3 (4) + 2 (0) = 12$ . Choice B has the intercepts reversed.
Question 7
A theater models its ticket sales for one show with $20 x + 50 y = 1, 000$ , where $x$ is the number of $20 tickets sold and $y$ is the number of $50 tickets sold. What is the best interpretation of the $x$ -intercept of the graph of this equation in the $x y$ -plane?
C. At the $x$ -intercept, $y = 0$ , so $20 x = 1, 000$ and $x = 50$ . That means 50 of the $20 tickets and none of the $50 tickets. Choice B describes the $y$ -intercept, (0, 20).
Question 8
In the $x y$ -plane, the graph of $c x + 3 y = 12$ passes through the point (2, 6), where $c$ is a constant. What is the value of $c$ ?
A. Substitute $x = 2$ and $y = 6$ : $2 c + 18 = 12$ . Then $2 c = - 6$ , so $c = - 3$ . Choice B comes from swapping the coordinates.
Question 9
$x$ $y$
0 7
2 3
5 −3
The table shows three points on a line in the $x y$ -plane. Which equation represents the line?
D. When $x$ increases by 2, $y$ decreases by 4, so the slope is −2 and the line is $y = - 2 x + 7$ . Adding $2 x$ to both sides gives $2 x + y = 7$ . Choice C passes through (0, 7) but has slope 2.
Question 10
Line $k$ is perpendicular to the graph of $2 x - 5 y = 10$ in the $x y$ -plane and passes through (0, 4). Which equation represents line $k$ ?
B. Solving $2 x - 5 y = 10$ for $y$ gives $y = 0.4 x - 2$ , so its slope is 0.4. A perpendicular line has slope $- \frac{1}{0.4} = - 2.5$ . With $y$ -intercept 4, line $k$ is $y = - 2.5 x + 4$ .
Question 11
In the $x y$ -plane, the graph of $3 x + k y = 24$ , where $k$ is a positive constant, has $x$ -intercept $(r, 0)$ and $y$ -intercept $(0, s)$ . If $r = 2 s$ , what is the value of $k$ ?
C. Setting $y = 0$ gives $3 r = 24$ , so $r = 8$ . Since $r = 2 s$ , $s = 4$ . Setting $x = 0$ gives $4 k = 24$ , so $k = 6$ . Choice A comes from making $s$ twice $r$ instead.
Question 12
A line in the $x y$ -plane has $x$ -intercept $(c, 0)$ and $y$ -intercept $(0, d)$ , where $c$ and $d$ are positive. What is the slope of the line?
B. From $(c, 0)$ to $(0, d)$ , $y$ changes by $d$ and $x$ changes by $- c$ . The slope is $\frac{d}{- c} = - \frac{d}{c}$ . With both intercepts positive, the line falls from left to right.

0/12 correct

Check your two-variable equation answers

Each row is a wrong answer choice from one of the sheet's problems. Find the one you picked, then run the check in the last column.

What your answer looked like	What actually happened	Fix it next time
$(0, 10)$ for the y-intercept of $4 x + 5 y = 40$ (problem 1)	10 comes from setting $y = 0$ , which gives the x-intercept, $(10, 0)$ .	For the y-intercept, set $x = 0$ : $5 y = 40$ , so the point is $(0, 8)$ .
$(0, 4)$ in problem 2	The minus sign got dropped. At $(0, 4)$ , $2 (0) - 3 (4) = - 12$ , not 12.	Substitute every choice with its signs. Only $(6, 0)$ works: $2 (6) - 3 (0) = 12$ .
$2 n + 3 f = 24$ in problem 3	Each price is attached to the wrong item. Notebooks cost 3 dollars each, so $n$ notebooks cost $3 n$ dollars.	Pair each price with its own variable: $3 n + 2 f = 24$ . Test a real purchase: 2 notebooks and 9 folders cost $3 (2) + 2 (9) = 24$ dollars, but $2 n + 3 f$ gives 31.
6 or 3 for the slope of $6 x + 2 y = 7$ (problem 5)	In standard form, the coefficient of $x$ isn't the slope. The slope is the coefficient of $x$ after you solve for $y$ .	$2 y = - 6 x + 7$ , so $y = - 3 x + \frac{7}{2}$ and the slope is $- 3$ .
1 in problem 8	You swapped the coordinates and substituted $x = 6$ and $y = 2$ .	In $(2, 6)$ , $x$ comes first: $2 c + 3 (6) = 12$ , so $2 c = - 6$ and $c = - 3$ .
$y = - 0.4 x + 4$ or $y = 2.5 x + 4$ in problem 10	A perpendicular slope takes two changes: flip the fraction and change the sign. Each of these choices makes only one.	The given line is $y = 0.4 x - 2$ , with slope $\frac{2}{5}$ . Flip and negate: $- \frac{5}{2} = - 2.5$ . Check: $0.4 \times (- 2.5) = - 1$ .

Worked example: the intercept question with a constant

Problem 11 gives $3 x + k y = 24$ with x-intercept $(r, 0)$ and y-intercept $(0, s)$ , where $r = 2 s$ . Start with the intercept you can find. Set $y = 0$ : $3 r = 24$ , so $r = 8$ . Then $r = 2 s$ makes $s = 4$ .

The point $(0, 4)$ is on the line, so substitute it: $3 (0) + k (4) = 24$ , which gives $k = 6$ . Check: $3 x + 6 y = 24$ crosses the axes at $(8, 0)$ and $(0, 4)$ , and 8 is twice 4. If you got 1.5, you made $s$ twice $r$ instead: $s = 16$ and $16 k = 24$ .

Apply the check to fresh questions

For a fresh two-variable equation, find both intercepts and substitute a point to check the signs. Use the practice link below for more standard-form and word-problem questions.

Practice two-variable equations questions

Review the linear equations in two variables method

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