# SAT Linear Equations in Two Variables Worksheet PDF with Answers Source: https://1600.now/sat-linear-equations-two-variables-worksheet --- ## 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 $y=mx+b$. Written by [Luke Finigan](https://1600.now/about) 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 [Download 4-page PDF](https://1600.now/downloads/worksheets/sat-linear-equations-two-variables-worksheet-student-letter.pdf) [Share to Google Classroom](https://classroom.google.com/share?url=https%3A%2F%2F1600.now%2Fsat-linear-equations-two-variables-worksheet) · This worksheet is free to print and share under [CC BY-NC-ND 4.0](https://creativecommons.org/licenses/by-nc-nd/4.0/). Tutors may also use it in paid sessions. If you post it online, please link to this page instead of re-uploading the PDF. These are original questions, not College Board items. ### 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 $B\ne0$, solving for $y$ gives $y=-\dfrac ABx+\dfrac CB$. The slope is $-\dfrac AB$, so the coefficient of $x$ in standard form cannot be read directly as the slope. If $B=0$ and $A\ne0$, 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 $3x+2y=12$? · A $3$ B $-\dfrac32$ C $\dfrac23$ ### Questions 1. **Question 1** What is the $y$-intercept of the graph of $4x + 5y = 40$ in the $xy$-plane? A. (0, 4) B. (0, 5) C. (0, 8) D. (0, 10) **C.** The $y$-intercept is where $x = 0$. Then $5y = 40$, so $y = 8$. The point $(10, 0)$ is the $x$-intercept. 2. **Question 2** Which point lies on the graph of $2x - 3y = 12$ in the $xy$-plane? A. (0, 4) B. (3, 2) C. (4, 6) D. (6, 0) **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 $(0, -4)$. 3. **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? A. $2n+3f=24$ B. $n+f=24$ C. $5(n+f)=24$ D. $3n+2f=24$ **D.** The notebooks cost $3n$ dollars and the folders cost $2f$ dollars. Together they cost 24 dollars, so $3n+2f=24$. Choice A attaches each price to the wrong item. 4. **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 $15t+10b=600$. If she tutored for 20 hours, how many hours did she work at the bookstore? A. 20 B. 30 C. 40 D. 60 **B.** Substitute $t=20$: $300+10b=600$, so $10b=300$ and $b=30$. Choice D, 60, is the number of bookstore hours if she had not tutored at all. 5. **Question 5** What is the slope of the graph of $6x+2y=7$ in the $xy$-plane? A. −3 B. $-\dfrac{1}{3}$ C. 3 D. 6 **A.** Solve for $y$: $2y=-6x+7$, so $y=-3x+\dfrac{7}{2}$. The slope is the coefficient of $x$, which is −3. 6. **Question 6** Which equation represents the line shown in the $xy$-plane? A. $3x+2y=12$ B. $2x+3y=12$ C. $2x+3y=18$ D. $3x-2y=12$ **A.** The line passes through (0, 6) and (4, 0). Both points satisfy $3x+2y=12$: $3(0)+2(6)=12$ and $3(4)+2(0)=12$. Choice B has the intercepts reversed. 7. **Question 7** A theater models its ticket sales for one show with $20x+50y=1{,}000$, 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 $xy$-plane? A. The theater's total ticket sales were $50. B. If no $20 tickets were sold, 20 of the $50 tickets were sold. C. If no $50 tickets were sold, 50 of the $20 tickets were sold. D. The theater could sell at most 50 of the $50 tickets. **C.** At the $x$-intercept, $y=0$, so $20x=1{,}000$ and $x=50$. That means 50 of the $20 tickets and none of the $50 tickets. Choice B describes the $y$-intercept, (0, 20). 8. **Question 8** In the $xy$-plane, the graph of $cx+3y=12$ passes through the point (2, 6), where $c$ is a constant. What is the value of $c$? A. −3 B. 1 C. 3 D. 15 **A.** Substitute $x=2$ and $y=6$: $2c+18=12$. Then $2c=-6$, so $c=-3$. Choice B comes from swapping the coordinates. 9. **Question 9** | $x$ | $y$ | | --- | --- | | 0 | 7 | | 2 | 3 | | 5 | −3 | The table shows three points on a line in the $xy$-plane. Which equation represents the line? A. $x+2y=7$ B. $2x-y=7$ C. $-2x+y=7$ D. $2x+y=7$ **D.** When $x$ increases by 2, $y$ decreases by 4, so the slope is −2 and the line is $y=-2x+7$. Adding $2x$ to both sides gives $2x+y=7$. Choice C passes through (0, 7) but has slope 2. 10. **Question 10** Line $k$ is perpendicular to the graph of $2x-5y=10$ in the $xy$-plane and passes through (0, 4). Which equation represents line $k$? A. $y=0.4x+4$ B. $y=-2.5x+4$ C. $y=2.5x+4$ D. $y=-0.4x+4$ **B.** Solving $2x-5y=10$ for $y$ gives $y=0.4x-2$, so its slope is 0.4. A perpendicular line has slope $-\dfrac{1}{0.4}=-2.5$. With $y$-intercept 4, line $k$ is $y=-2.5x+4$. 11. **Question 11** In the $xy$-plane, the graph of $3x+ky=24$, where $k$ is a positive constant, has $x$-intercept $(r,0)$ and $y$-intercept $(0,s)$. If $r=2s$, what is the value of $k$? A. 1.5 B. 3 C. 6 D. 12 **C.** Setting $y=0$ gives $3r=24$, so $r=8$. Since $r=2s$, $s=4$. Setting $x=0$ gives $4k=24$, so $k=6$. Choice A comes from making $s$ twice $r$ instead. 12. **Question 12** A line in the $xy$-plane has $x$-intercept $(c,0)$ and $y$-intercept $(0,d)$, where $c$ and $d$ are positive. What is the slope of the line? A. $-\dfrac{c}{d}$ B. $-\dfrac{d}{c}$ C. $\dfrac{c}{d}$ D. $\dfrac{d}{c}$ **B.** From $(c,0)$ to $(0,d)$, $y$ changes by $d$ and $x$ changes by $-c$. The slope is $\dfrac{d}{-c}=-\dfrac{d}{c}$. 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 $4x + 5y = 40$ (problem 1) | 10 comes from setting $y = 0$, which gives the x-intercept, $(10, 0)$. | For the y-intercept, set $x = 0$: $5y = 40$, 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: $2(6) - 3(0) = 12$. | | $2n + 3f = 24$ in problem 3 | Each price is attached to the wrong item. Notebooks cost 3 dollars each, so $n$ notebooks cost $3n$ dollars. | Pair each price with its own variable: $3n + 2f = 24$. Test a real purchase: 2 notebooks and 9 folders cost $3(2) + 2(9) = 24$ dollars, but $2n + 3f$ gives 31. | | 6 or 3 for the slope of $6x + 2y = 7$ (problem 5) | In standard form, the coefficient of $x$ isn't the slope. The slope is the coefficient of $x$ after you solve for $y$. | $2y = -6x + 7$, so $y = -3x + \dfrac{7}{2}$ 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: $2c + 3(6) = 12$, so $2c = -6$ and $c = -3$. | | $y = -0.4x + 4$ or $y = 2.5x + 4$ 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 $y = 0.4x - 2$, with slope $\dfrac{2}{5}$. Flip and negate: $-\dfrac{5}{2} = -2.5$. Check: $0.4 \times (-2.5) = -1$. | #### Worked example: the intercept question with a constant Problem 11 gives $3x + ky = 24$ with x-intercept $(r, 0)$ and y-intercept $(0, s)$, where $r = 2s$. Start with the intercept you can find. Set $y = 0$: $3r = 24$, so $r = 8$. Then $r = 2s$ makes $s = 4$. The point $(0, 4)$ is on the line, so substitute it: $3(0) + k(4) = 24$, which gives $k = 6$. Check: $3x + 6y = 24$ 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 $16k = 24$. ### 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](https://1600.now/bank/math/skill/Linear%20equations%20in%20two%20variables) [Review the linear equations in two variables method](https://1600.now/sat-skill/linear-equations-two-variables) ### More Math worksheets [All worksheets](https://1600.now/sat-worksheets) - [Linear Equations](https://1600.now/sat-linear-equations-one-variable-worksheet) - [Systems of Equations](https://1600.now/sat-systems-of-linear-equations-worksheet) - [Linear Functions](https://1600.now/sat-linear-functions-worksheet)