# Exercises: Solve Systems of Equations by Substitution (4.2)

**Exercises: Solve a System of Equations by Substitution**

Instructions: For questions 1-36, solve the systems of equations by substitution.

**1. [latex]\left\{\begin{array}{c}2x+y=-4\\ 3x-2y=-6\end{array}\right.[/latex]**

**Solution**

[latex]\left(-2,0\right)[/latex]

**2. [latex]\left\{\begin{array}{c}2x+y=-2\\ 3x-y=7\end{array}\right.[/latex]**

**3. [latex]\left\{\begin{array}{c}x-2y=-5\\ 2x-3y=-4\end{array}\right.[/latex]**

**Solution**

[latex]\left(7,6\right)[/latex]

**4. [latex]\left\{\begin{array}{c}x-3y=-9\\ 2x+5y=4\end{array}\right.[/latex]**

**5. [latex]\left\{\begin{array}{c}5x-2y=-6\\ y=3x+3\end{array}\right.[/latex]**

**Solution**

[latex]\left(0,3\right)[/latex]

**6. [latex]\left\{\begin{array}{c}-2x+2y=6\\ y=-3x+1\end{array}\right.[/latex]**

**7. [latex]\left\{\begin{array}{c}2x+3y=3\\ y=\text{−}x+3\end{array}\right.[/latex]**

**Solution**

[latex]\left(6,-3\right)[/latex]

**8. [latex]\left\{\begin{array}{c}2x+5y=-14\\ y=-2x+2\end{array}\right.[/latex]**

**9. [latex]\left\{\begin{array}{c}2x+5y=1\\ y=\frac{1}{3}x-2\end{array}\right.[/latex]**

**Solution**

[latex]\left(3,-1\right)[/latex]

**10. [latex]\left\{\begin{array}{c}3x+4y=1\\ y=-\frac{2}{5}x+2\end{array}\right.[/latex]**

**11. [latex]\left\{\begin{array}{c}3x-2y=6\\ y=\frac{2}{3}x+2\end{array}\right.[/latex]**

**Solution**

[latex]\left(6,6\right)[/latex]

**12. [latex]\left\{\begin{array}{c}-3x-5y=3\\ y=\frac{1}{2}x-5\end{array}\right.[/latex]**

**13. [latex]\left\{\begin{array}{c}2x+y=10\\ -x+y=-5\end{array}\right.[/latex]**

**Solution**

[latex]\left(5,0\right)[/latex]

**14. [latex]\left\{\begin{array}{c}-2x+y=10\\ -x+2y=16\end{array}\right.[/latex]**

**15. [latex]\left\{\begin{array}{c}3x+y=1\\ -4x+y=15\end{array}\right.[/latex]**

**Solution**

[latex]\left(-2,7\right)[/latex]

**16. [latex]\left\{\begin{array}{c}x+y=0\\ 2x+3y=-4\end{array}\right.[/latex]**

**17. [latex]\left\{\begin{array}{c}x+3y=1\\ 3x+5y=-5\end{array}\right.[/latex]**

**Solution**

[latex]\left(-5,2\right)[/latex]

**18. [latex]\left\{\begin{array}{c}x+2y=-1\\ 2x+3y=1\end{array}\right.[/latex]**

**19. [latex]\left\{\begin{array}{c}2x+y=5\\ x-2y=-15\end{array}\right.[/latex]**

**Solution**

[latex]\left(-1,7\right)[/latex]

**20. [latex]\left\{\begin{array}{c}4x+y=10\\ x-2y=-20\end{array}\right.[/latex]**

**21. [latex]\left\{\begin{array}{c}y=-2x-1\\ y=-\frac{1}{3}x+4\end{array}\right.[/latex]**

**Solution**

[latex]\left(-3,5\right)[/latex]

**22. [latex]\left\{\begin{array}{c}y=x-6\\ y=-\frac{3}{2}x+4\end{array}\right.[/latex]**

**23. [latex]\left\{\begin{array}{c}y=2x-8\\ y=\frac{3}{5}x+6\end{array}\right.[/latex]**

**Solution**

(10, 12)

**24. [latex]\left\{\begin{array}{c}y=\text{−}x-1\\ y=x+7\end{array}\right.[/latex]**

**25. [latex]\left\{\begin{array}{c}4x+2y=8\\ 8x-y=1\end{array}\right.[/latex]**

**Solution**

[latex]\left(\frac{1}{2},3\right)[/latex]

**26. [latex]\left\{\begin{array}{c}-x-12y=-1\\ 2x-8y=-6\end{array}\right.[/latex]**

**27. [latex]\left\{\begin{array}{c}15x+2y=6\\ -5x+2y=-4\end{array}\right.[/latex]**

**Solution**

[latex]\left(\frac{1}{2},-\frac{3}{4}\right)[/latex]

**28. [latex]\left\{\begin{array}{c}2x-15y=7\\ 12x+2y=-4\end{array}\right.[/latex]**

**29. [latex]\left\{\begin{array}{c}y=3x\\ 6x-2y=0\end{array}\right.[/latex]**

**Solution**

Infinitely many solutions

**30. [latex]\left\{\begin{array}{c}x=2y\\ 4x-8y=0\end{array}\right.[/latex]**

**31. [latex]\left\{\begin{array}{c}2x+16y=8\\ -x-8y=-4\end{array}\right.[/latex]**

**Solution**

Infinitely many solutions

**32. [latex]\left\{\begin{array}{c}15x+4y=6\\ -30x-8y=-12\end{array}\right.[/latex]**

**33. [latex]\left\{\begin{array}{c}y=-4x\\ 4x+y=1\end{array}\right.[/latex]**

**Solution**

No solution

**34. [latex]\left\{\begin{array}{c}y=-\frac{1}{4}x\\ x+4y=8\end{array}\right.[/latex]**

**35. [latex]\left\{\begin{array}{c}y=\frac{7}{8}x+4\\ -7x+8y=6\end{array}\right.[/latex]**

**Solution**

No solution

**36. [latex]\left\{\begin{array}{c}y=-\frac{2}{3}x+5\\ 2x+3y=11\end{array}\right.[/latex]**

**Exercises: Solve Applications of Systems of Equations by Substitution**

Instructions: For questions 37-51, translate to a system of equations and solve.

**37. The sum of two numbers is [latex]15[/latex]. One number is [latex]3[/latex] less than the other. Find the numbers.**

**Solution**

The numbers are [latex]6[/latex] and [latex]9[/latex].

**38. The sum of two numbers is [latex]30[/latex]. One number is [latex]4[/latex] less than the other. Find the numbers.**

**39. The sum of two numbers is [latex]-26[/latex]. One number is [latex]12[/latex] less than the other. Find the numbers.**

**Solution**

The numbers are [latex]-7[/latex] and [latex]-19[/latex].

**40. The perimeter of a rectangle is [latex]50[/latex]. The length is [latex]5[/latex] more than the width. Find the length and width.**

**41. The perimeter of a rectangle is [latex]60[/latex]. The length is [latex]10[/latex] more than the width. Find the length and width.**

**Solution**

The length is [latex]20[/latex] and the width is [latex]10[/latex].

**42. The perimeter of a rectangle is [latex]58[/latex]. The length is [latex]5[/latex] more than three times the width. Find the length and width.**

**43. The perimeter of a rectangle is [latex]84[/latex]. The length is [latex]10[/latex] more than three times the width. Find the length and width.**

**Solution**

The length is [latex]34[/latex] and the width is [latex]8[/latex].

**44. The measure of one of the small angles of a right triangle is [latex]14[/latex] more than [latex]3[/latex] times the measure of the other small angle. Find the measure of both angles.**

**45. The measure of one of the small angles of a right triangle is [latex]26[/latex] more than [latex]3[/latex] times the measure of the other small angle. Find the measure of both angles.**

**Solution**

The measures are [latex]16^\circ[/latex] and [latex]74^\circ[/latex].

**46. The measure of one of the small angles of a right triangle is [latex]15[/latex] less than twice the measure of the other small angle. Find the measure of both angles.**

**47. The measure of one of the small angles of a right triangle is [latex]45[/latex] less than twice the measure of the other small angle. Find the measure of both angles.**

**Solution**

The measures are [latex]45^\circ[/latex] and [latex]45^\circ[/latex].

**48. Maxim has been offered positions by two car dealers. The first company pays a salary of [latex]$10\text{,}000[/latex] plus a commission of [latex]$1\text{,}000[/latex] for each car sold. The second pays a salary of [latex]$20\text{,}000[/latex] plus a commission of [latex]$500[/latex] for each car sold. How many cars would need to be sold to make the total pay the same?**

**49. Jackie has been offered positions by two cable companies. The first company pays a salary of [latex]14\text{,}000[/latex] plus a commission of [latex]$100[/latex] for each cable package sold. The second pays a salary of [latex]$20\text{,}000[/latex] plus a commission of [latex]$25[/latex] for each cable package sold. How many cable packages would need to be sold to make the total pay the same?**

**Solution**

[latex]80[/latex] cable packages would need to be sold.

**50. Amara currently sells televisions for company. A at a salary of [latex]$17\text{,}000[/latex] plus a [latex]$100[/latex] commission for each television she sells. Company B offers her a position with a salary of [latex]$29\text{,}000[/latex] plus a [latex]$20[/latex] commission for each television she sells. How televisions would Amara need to sell for the options to be equal?**

**51. Mitchell currently sells stoves for company. A at a salary of [latex]$12\text{,}000[/latex] plus a [latex]$150[/latex] commission for each stove he sells. Company B offers him a position with a salary of [latex]$24\text{,}000[/latex] plus a [latex]$50[/latex] commission for each stove he sells. How many stoves would Mitchell need to sell for the options to be equal?**

**Solution**

Mitchell would need to sell [latex]120[/latex] stoves.

**Exercises: Everyday Math**

**52. When Gloria spent [latex]15[/latex] minutes on the elliptical trainer and then did circuit training for [latex]30[/latex] minutes, her fitness app says she burned [latex]435[/latex] calories. When she spent [latex]30[/latex] minutes on the elliptical trainer and [latex]40[/latex] minutes circuit training she burned [latex]690[/latex] calories. Solve the system [latex]\left\{\begin{array}{c}15e+30c=435\\ 30e+40c=690\end{array}\right.[/latex] for [latex]e[/latex], the number of calories she burns for each minute on the elliptical trainer, and [latex]c[/latex], the number of calories she burns for each minute of circuit training.**

**53. Stephanie left Riverside, California, driving her motorhome north on Interstate 15 towards Salt Lake City at a speed of [latex]56[/latex] miles per hour. Half an hour later, Tina left Riverside in her car on the same route as Stephanie, driving [latex]70[/latex] miles per hour. Solve the system [latex]\left\{\begin{array}{c}56s=70t\\ s=t+\frac{1}{2}\end{array}\right.[/latex].**

**a. for [latex]t[/latex] to find out how long it will take Tina to catch up to Stephanie.**

**b. what is the value of [latex]s[/latex], the number of hours Stephanie will have driven before Tina catches up to her?**

**Solution**

a.[latex]t=2[/latex] hours

b. [latex]s=2\frac{1}{2}[/latex] hours

**Exercises: Writing Exercises**

**54. Solve the system of equations ****[latex]\left\{\begin{array}{c}x+y=10\\ x-y=6\end{array}\right.[/latex]**

**a. by graphing
**

**b. by substitution**

**c. Which method do you prefer? Why?**

**55. Solve the system of equations [latex]\left\{\begin{array}{c}3x+y=12\\ x=y-8\end{array}\right.[/latex] by substitution and explain all your steps in words.**

**Solution**

Answers will vary.