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. $\left\{\begin{array}{c}2x+y=-4\\ 3x-2y=-6\end{array}\right.$
Solution

$\left(-2,0\right)$

2. $\left\{\begin{array}{c}2x+y=-2\\ 3x-y=7\end{array}\right.$

3. $\left\{\begin{array}{c}x-2y=-5\\ 2x-3y=-4\end{array}\right.$
Solution

$\left(7,6\right)$

4. $\left\{\begin{array}{c}x-3y=-9\\ 2x+5y=4\end{array}\right.$

5. $\left\{\begin{array}{c}5x-2y=-6\\ y=3x+3\end{array}\right.$
Solution

$\left(0,3\right)$

6. $\left\{\begin{array}{c}-2x+2y=6\\ y=-3x+1\end{array}\right.$

7. $\left\{\begin{array}{c}2x+3y=3\\ y=\text{−}x+3\end{array}\right.$
Solution

$\left(6,-3\right)$

8. $\left\{\begin{array}{c}2x+5y=-14\\ y=-2x+2\end{array}\right.$

9. $\left\{\begin{array}{c}2x+5y=1\\ y=\frac{1}{3}x-2\end{array}\right.$
Solution

$\left(3,-1\right)$

10. $\left\{\begin{array}{c}3x+4y=1\\ y=-\frac{2}{5}x+2\end{array}\right.$

11. $\left\{\begin{array}{c}3x-2y=6\\ y=\frac{2}{3}x+2\end{array}\right.$
Solution

$\left(6,6\right)$

12. $\left\{\begin{array}{c}-3x-5y=3\\ y=\frac{1}{2}x-5\end{array}\right.$

13. $\left\{\begin{array}{c}2x+y=10\\ -x+y=-5\end{array}\right.$
Solution

$\left(5,0\right)$

14. $\left\{\begin{array}{c}-2x+y=10\\ -x+2y=16\end{array}\right.$

15. $\left\{\begin{array}{c}3x+y=1\\ -4x+y=15\end{array}\right.$
Solution

$\left(-2,7\right)$

16. $\left\{\begin{array}{c}x+y=0\\ 2x+3y=-4\end{array}\right.$

17. $\left\{\begin{array}{c}x+3y=1\\ 3x+5y=-5\end{array}\right.$
Solution

$\left(-5,2\right)$

18. $\left\{\begin{array}{c}x+2y=-1\\ 2x+3y=1\end{array}\right.$

19. $\left\{\begin{array}{c}2x+y=5\\ x-2y=-15\end{array}\right.$
Solution

$\left(-1,7\right)$

20. $\left\{\begin{array}{c}4x+y=10\\ x-2y=-20\end{array}\right.$

21. $\left\{\begin{array}{c}y=-2x-1\\ y=-\frac{1}{3}x+4\end{array}\right.$
Solution

$\left(-3,5\right)$

22. $\left\{\begin{array}{c}y=x-6\\ y=-\frac{3}{2}x+4\end{array}\right.$

23. $\left\{\begin{array}{c}y=2x-8\\ y=\frac{3}{5}x+6\end{array}\right.$
Solution

(10, 12)

24. $\left\{\begin{array}{c}y=\text{−}x-1\\ y=x+7\end{array}\right.$

25. $\left\{\begin{array}{c}4x+2y=8\\ 8x-y=1\end{array}\right.$
Solution

$\left(\frac{1}{2},3\right)$

26. $\left\{\begin{array}{c}-x-12y=-1\\ 2x-8y=-6\end{array}\right.$

27. $\left\{\begin{array}{c}15x+2y=6\\ -5x+2y=-4\end{array}\right.$
Solution

$\left(\frac{1}{2},-\frac{3}{4}\right)$

28. $\left\{\begin{array}{c}2x-15y=7\\ 12x+2y=-4\end{array}\right.$

29. $\left\{\begin{array}{c}y=3x\\ 6x-2y=0\end{array}\right.$
Solution

Infinitely many solutions

30. $\left\{\begin{array}{c}x=2y\\ 4x-8y=0\end{array}\right.$

31. $\left\{\begin{array}{c}2x+16y=8\\ -x-8y=-4\end{array}\right.$
Solution

Infinitely many solutions

32. $\left\{\begin{array}{c}15x+4y=6\\ -30x-8y=-12\end{array}\right.$

33. $\left\{\begin{array}{c}y=-4x\\ 4x+y=1\end{array}\right.$
Solution

No solution

34. $\left\{\begin{array}{c}y=-\frac{1}{4}x\\ x+4y=8\end{array}\right.$

35. $\left\{\begin{array}{c}y=\frac{7}{8}x+4\\ -7x+8y=6\end{array}\right.$
Solution

No solution

36. $\left\{\begin{array}{c}y=-\frac{2}{3}x+5\\ 2x+3y=11\end{array}\right.$

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 $15$. One number is $3$ less than the other. Find the numbers.
Solution

The numbers are $6$ and $9$.

38. The sum of two numbers is $30$. One number is $4$ less than the other. Find the numbers.

39. The sum of two numbers is $-26$. One number is $12$ less than the other. Find the numbers.
Solution

The numbers are $-7$ and $-19$.

40. The perimeter of a rectangle is $50$. The length is $5$ more than the width. Find the length and width.

41. The perimeter of a rectangle is $60$. The length is $10$ more than the width. Find the length and width.
Solution

The length is $20$ and the width is $10$.

42. The perimeter of a rectangle is $58$. The length is $5$ more than three times the width. Find the length and width.

43. The perimeter of a rectangle is $84$. The length is $10$ more than three times the width. Find the length and width.
Solution

The length is $34$ and the width is $8$.

44. The measure of one of the small angles of a right triangle is $14$ more than $3$ 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 $26$ more than $3$ times the measure of the other small angle. Find the measure of both angles.
Solution

The measures are $16^\circ$ and $74^\circ$.

46. The measure of one of the small angles of a right triangle is $15$ 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 $45$ less than twice the measure of the other small angle. Find the measure of both angles.
Solution

The measures are $45^\circ$ and $45^\circ$.

48. Maxim has been offered positions by two car dealers. The first company pays a salary of $10\text{,}000$ plus a commission of $1\text{,}000$ for each car sold. The second pays a salary of $20\text{,}000$ plus a commission of $500$ 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 $14\text{,}000$ plus a commission of $100$ for each cable package sold. The second pays a salary of $20\text{,}000$ plus a commission of $25$ for each cable package sold. How many cable packages would need to be sold to make the total pay the same?
Solution

$80$ cable packages would need to be sold.

50. Amara currently sells televisions for company. A at a salary of $17\text{,}000$ plus a $100$ commission for each television she sells. Company B offers her a position with a salary of $29\text{,}000$ plus a $20$ 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 $12\text{,}000$ plus a $150$ commission for each stove he sells. Company B offers him a position with a salary of $24\text{,}000$ plus a $50$ 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 $120$ stoves.

Exercises: Everyday Math

Instructions: For questions 52-53, answer the given everyday math word problems.
52. When Gloria spent $15$ minutes on the elliptical trainer and then did circuit training for $30$ minutes, her fitness app says she burned $435$ calories. When she spent $30$ minutes on the elliptical trainer and $40$ minutes circuit training she burned $690$ calories. Solve the system $\left\{\begin{array}{c}15e+30c=435\\ 30e+40c=690\end{array}\right.$ for $e$, the number of calories she burns for each minute on the elliptical trainer, and $c$, 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 $56$ miles per hour. Half an hour later, Tina left Riverside in her car on the same route as Stephanie, driving $70$ miles per hour. Solve the system $\left\{\begin{array}{c}56s=70t\\ s=t+\frac{1}{2}\end{array}\right.$.

a. for $t$ to find out how long it will take Tina to catch up to Stephanie.
b. what is the value of $s$, the number of hours Stephanie will have driven before Tina catches up to her?

Solution

a.$t=2$ hours
b. $s=2\frac{1}{2}$ hours

Exercises: Writing Exercises

Instructions: For questions 54-55, answer the given writing exercises.

54. Solve the system of equations $\left\{\begin{array}{c}x+y=10\\ x-y=6\end{array}\right.$

a. by graphing
b. by substitution
c. Which method do you prefer? Why?

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