# Exercises: Solve Systems of Equations by Elimination (4.3)

## Exercises: Solve a System of Equations by Elimination

Instructions: For questions 1-41, solve the systems of equations by elimination.
1. $\left\{\begin{array}{c}5x+2y=2\\ -3x-y=0\end{array}\right.$

2. $\left\{\begin{array}{c}-3x+y=-9\\ x-2y=-12\end{array}\right.$
Solution

$(6,9)$

3. $\left\{\begin{array}{c}6x-5y=-1\\ 2x+y=13\end{array}\right.$

4. $\left\{\begin{array}{c}3x-y=-7\\ 4x+2y=-6\end{array}\right.$
Solution

$\left(-2,1\right)$

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

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

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

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

8. $\left\{\begin{array}{c}-7x+6y=-10\\ x-6y=22\end{array}\right.$
Solution

$\left(-2,-4\right)$

9. $\left\{\begin{array}{c}3x+2y=-3\\ -x-2y=-19\end{array}\right.$

10. $\left\{\begin{array}{c}5x+2y=1\\ -5x-4y=-7\end{array}\right.$
Solution

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

11. $\left\{\begin{array}{c}6x+4y=-4\\ -6x-5y=8\end{array}\right.$

12. $\left\{\begin{array}{c}3x-4y=-11\\ x-2y=-5\end{array}\right.$
Solution

$\left(-1,2\right)$

13. $\left\{\begin{array}{c}5x-7y=29\\ x+3y=-3\end{array}\right.$

14. $\left\{\begin{array}{c}6x-5y=-75\\ -x-2y=-13\end{array}\right.$
Solution

$\left(-5,9\right)$

15. $\left\{\begin{array}{c}-x+4y=8\\ 3x+5y=10\end{array}\right.$

16. $\left\{\begin{array}{c}2x-5y=7\\ 3x-y=17\end{array}\right.$
Solution

$(6,1)$

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

18. $\left\{\begin{array}{c}7x+y=-4\\ 13x+3y=4\end{array}\right.$
Solution

$\left(-2,10\right)$

19. $\left\{\begin{array}{c}-3x+5y=-13\\ 2x+y=-26\end{array}\right.$

20. $\left\{\begin{array}{c}3x-5y=-9\\ 5x+2y=16\end{array}\right.$
Solution

$(2,3)$

21. $\left\{\begin{array}{c}4x-3y=3\\ 2x+5y=-31\end{array}\right.$

22. $\left\{\begin{array}{c}4x+7y=14\\ -2x+3y=32\end{array}\right.$
Solution

$\left(-7,6\right)$

23. $\left\{\begin{array}{c}5x+2y=21\\ 7x-4y=9\end{array}\right.$

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

$\left(-9,3\right)$

25. $\left\{\begin{array}{c}11x+9y=-5\\ 7x+5y=-1\end{array}\right.$

26. $\left\{\begin{array}{c}3x+8y=67\\ 5x+3y=60\end{array}\right.$
Solution

$(9,5)$

27. $\left\{\begin{array}{c}2x+9y=-4\\ 3x+13y=-7\end{array}\right.$

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

$\left(-3,2\right)$

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

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

$\left(-2,3\right)$

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

32. $\left\{\begin{array}{c}2x+y=3\\ 6x+3y=9\end{array}\right.$
Solution

infinitely many solutions

33. $\left\{\begin{array}{c}x-4y=-1\\ -3x+12y=3\end{array}\right.$

34. $\left\{\begin{array}{c}-3x-y=8\\ 6x+2y=-16\end{array}\right.$
Solution

infinitely many solutions

35. $\left\{\begin{array}{c}4x+3y=2\\ 20x+15y=10\end{array}\right.$

36. $\left\{\begin{array}{c}3x+2y=6\\ -6x-4y=-12\end{array}\right.$
Solution

infinitely many solutions

37. $\left\{\begin{array}{c}5x-8y=12\\ 10x-16y=20\end{array}\right.$

38. $\left\{\begin{array}{c}-11x+12y=60\\ -22x+24y=90\end{array}\right.$
Solution

inconsistent, no solution

39. $\left\{\begin{array}{c}7x-9y=16\\ -21x+27y=-24\end{array}\right.$

40. $\left\{\begin{array}{c}5x-3y=15\\ y=\frac{5}{3}x-2\end{array}\right.$
Solution

inconsistent, no solution

41. $\left\{\begin{array}{c}2x+4y=7\\ y=-\frac{1}{2}x-4\end{array}\right.$

## Exercises: Solve Applications of Systems of Equations by Elimination

Instructions: For questions 42-49, translate to a system of equations and solve.

42. The sum of two numbers is $65$. Their difference is $25$. Find the numbers.
Solution

The numbers are $20$ and $45$.

43. The sum of two numbers is $37$. Their difference is $9$. Find the numbers.

44. The sum of two numbers is $-27$. Their difference is $-59$. Find the numbers.
Solution

The numbers are $16$ and $-43$.

45. The sum of two numbers is $-45$. Their difference is $-89$. Find the numbers.

46. Andrea is buying some new shirts and sweaters. She is able to buy $3$ shirts and $2$ sweaters for $114$ or she is able to buy $2$ shirts and $4$ sweaters for $164$. How much does a shirt cost? How much does a sweater cost?
Solution

A shirt costs $16$ and a sweater costs $33$.

47. Peter is buying office supplies. He is able to buy $3$ packages of paper and $4$ staplers for $40$ or he is able to buy $5$ packages of paper and $6$ staplers for $62$. How much does a package of paper cost? How much does a stapler cost?

48. The total amount of sodium in $2$ hot dogs and $3$ cups of cottage cheese is $4720$ mg. The total amount of sodium in $5$ hot dogs and $2$ cups of cottage cheese is $6300$ mg. How much sodium is in a hot dog? How much sodium is in a cup of cottage cheese?
Solution

There are $860$ mg in a hot dog. There are $1\text{,}000$ mg in a cup of cottage cheese.

49. The total number of calories in $2$ hot dogs and $3$ cups of cottage cheese is $960$ calories. The total number of calories in $5$ hot dogs and $2$ cups of cottage cheese is $1190$ calories. How many calories are in a hot dog? How many calories are in a cup of cottage cheese?

## Exercises: Choose the Most Convenient Method to Solve a System of Linear Equations

Instructions: For questions 50-53, decide whether it would be more convenient to solve the system of equations by substitution or elimination.

50.

a.$\left\{\begin{array}{c}8x-15y=-32\\ 6x+3y=-5\end{array}\right.$
b.$\left\{\begin{array}{c}x=4y-3\\ 4x-2y=-6\end{array}\right.$

Solution

a. elimination
b. substitution

51.

a.$\left\{\begin{array}{c}y=7x-5\\ 3x-2y=16\end{array}\right.$
b.$\left\{\begin{array}{c}12x-5y=-42\\ 3x+7y=-15\end{array}\right.$

52.

a.$\left\{\begin{array}{c}y=4x+9\\ 5x-2y=-21\end{array}\right.$
b.$\left\{\begin{array}{c}9x-4y=24\\ 3x+5y=-14\end{array}\right.$

Solution

a. substitution
b. elimination

53.

a.$\left\{\begin{array}{c}14x-15y=-30\\ 7x+2y=10\end{array}\right.$
b.$\left\{\begin{array}{c}x=9y-11\\ 2x-7y=-27\end{array}\right.$

## Exercises: Everyday Math

Instructions: For questions 54-55, answer the given everyday math word problems.

54. Norris can row $3$ miles upstream against the current in the same amount of time it takes him to row $5$ miles downstream, with the current. Solve the system. $\left\{\begin{array}{c}r-c=3\\ r+c=5\end{array}\right.$

a. for $r$, his rowing speed in still water.
b. Then solve for $c$, the speed of the river current.

Solution

a. $r=4$
b. $c=1$

55. Josie wants to make $10$ pounds of trail mix using nuts and raisins, and she wants the total cost of the trail mix to be $54$. Nuts cost $6$ per pound and raisins cost $3$ per pound. Solve the system $\left\{\begin{array}{c}n+r=10\\ 6n+3r=54\end{array}\right.$ to find $n$, the number of pounds of nuts, and $r$, the number of pounds of raisins she should use.

## Exercises: Writing Exercises

Instructions: For questions 56-57, answer the given writing exercises.

56. Solve the system $\left\{\begin{array}{c}x+y=10\\ 5x+8y=56\end{array}\right.$

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

Solution

a. (8, 2)
b.

57. Solve the system $\left\{\begin{array}{c}x+y=-12\\ y=4-\frac{1}{2}x\end{array}\right.$