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. [latex]\left\{\begin{array}{c}5x+2y=2\\ -3x-y=0\end{array}\right.[/latex]
2. [latex]\left\{\begin{array}{c}-3x+y=-9\\ x-2y=-12\end{array}\right.[/latex]
Solution

[latex](6,9)[/latex]

3. [latex]\left\{\begin{array}{c}6x-5y=-1\\ 2x+y=13\end{array}\right.[/latex]
4. [latex]\left\{\begin{array}{c}3x-y=-7\\ 4x+2y=-6\end{array}\right.[/latex]
Solution

[latex]\left(-2,1\right)[/latex]

5. [latex]\left\{\begin{array}{c}x+y=-1\\ x-y=-5\end{array}\right.[/latex]
6. [latex]\left\{\begin{array}{c}x+y=-8\\ x-y=-6\end{array}\right.[/latex]
Solution

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

7. [latex]\left\{\begin{array}{c}3x-2y=1\\ -x+2y=9\end{array}\right.[/latex]
8. [latex]\left\{\begin{array}{c}-7x+6y=-10\\ x-6y=22\end{array}\right.[/latex]
Solution

[latex]\left(-2,-4\right)[/latex]

9. [latex]\left\{\begin{array}{c}3x+2y=-3\\ -x-2y=-19\end{array}\right.[/latex]
10. [latex]\left\{\begin{array}{c}5x+2y=1\\ -5x-4y=-7\end{array}\right.[/latex]
Solution

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

11. [latex]\left\{\begin{array}{c}6x+4y=-4\\ -6x-5y=8\end{array}\right.[/latex]
12. [latex]\left\{\begin{array}{c}3x-4y=-11\\ x-2y=-5\end{array}\right.[/latex]
Solution

[latex]\left(-1,2\right)[/latex]

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

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

[latex]\left(-5,9\right)[/latex]

15. [latex]\left\{\begin{array}{c}-x+4y=8\\ 3x+5y=10\end{array}\right.[/latex]
16. [latex]\left\{\begin{array}{c}2x-5y=7\\ 3x-y=17\end{array}\right.[/latex]
Solution

[latex](6,1)[/latex]

17. [latex]\left\{\begin{array}{c}5x-3y=-1\\ 2x-y=2\end{array}\right.[/latex]
18. [latex]\left\{\begin{array}{c}7x+y=-4\\ 13x+3y=4\end{array}\right.[/latex]
Solution

[latex]\left(-2,10\right)[/latex]

19. [latex]\left\{\begin{array}{c}-3x+5y=-13\\ 2x+y=-26\end{array}\right.[/latex]
20. [latex]\left\{\begin{array}{c}3x-5y=-9\\ 5x+2y=16\end{array}\right.[/latex]
Solution

[latex](2,3)[/latex]

21. [latex]\left\{\begin{array}{c}4x-3y=3\\ 2x+5y=-31\end{array}\right.[/latex]
22. [latex]\left\{\begin{array}{c}4x+7y=14\\ -2x+3y=32\end{array}\right.[/latex]
Solution

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

23. [latex]\left\{\begin{array}{c}5x+2y=21\\ 7x-4y=9\end{array}\right.[/latex]
24. [latex]\left\{\begin{array}{c}3x+8y=-3\\ 2x+5y=-3\end{array}\right.[/latex]
Solution

[latex]\left(-9,3\right)[/latex]

25. [latex]\left\{\begin{array}{c}11x+9y=-5\\ 7x+5y=-1\end{array}\right.[/latex]
26. [latex]\left\{\begin{array}{c}3x+8y=67\\ 5x+3y=60\end{array}\right.[/latex]
Solution

[latex](9,5)[/latex]

27. [latex]\left\{\begin{array}{c}2x+9y=-4\\ 3x+13y=-7\end{array}\right.[/latex]
28. [latex]\left\{\begin{array}{c}\frac{1}{3}x-y=-3\\ x+\frac{5}{2}y=2\end{array}\right.[/latex]
Solution

[latex]\left(-3,2\right)[/latex]

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

[latex]\left(-2,3\right)[/latex]

31. [latex]\left\{\begin{array}{c}\frac{1}{3}x-y=-3\\ \frac{2}{3}x+\frac{5}{2}y=3\end{array}\right.[/latex]
32. [latex]\left\{\begin{array}{c}2x+y=3\\ 6x+3y=9\end{array}\right.[/latex]
Solution

infinitely many solutions

33. [latex]\left\{\begin{array}{c}x-4y=-1\\ -3x+12y=3\end{array}\right.[/latex]
34. [latex]\left\{\begin{array}{c}-3x-y=8\\ 6x+2y=-16\end{array}\right.[/latex]
Solution

infinitely many solutions

35. [latex]\left\{\begin{array}{c}4x+3y=2\\ 20x+15y=10\end{array}\right.[/latex]
36. [latex]\left\{\begin{array}{c}3x+2y=6\\ -6x-4y=-12\end{array}\right.[/latex]
Solution

infinitely many solutions

37. [latex]\left\{\begin{array}{c}5x-8y=12\\ 10x-16y=20\end{array}\right.[/latex]
38. [latex]\left\{\begin{array}{c}-11x+12y=60\\ -22x+24y=90\end{array}\right.[/latex]
Solution

inconsistent, no solution

39. [latex]\left\{\begin{array}{c}7x-9y=16\\ -21x+27y=-24\end{array}\right.[/latex]
40. [latex]\left\{\begin{array}{c}5x-3y=15\\ y=\frac{5}{3}x-2\end{array}\right.[/latex]
Solution

inconsistent, no solution

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

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 [latex]65[/latex]. Their difference is [latex]25[/latex]. Find the numbers.
Solution

The numbers are [latex]20[/latex] and [latex]45[/latex].

43. The sum of two numbers is [latex]37[/latex]. Their difference is [latex]9[/latex]. Find the numbers.
44. The sum of two numbers is [latex]-27[/latex]. Their difference is [latex]-59[/latex]. Find the numbers.
Solution

The numbers are [latex]16[/latex] and [latex]-43[/latex].

45. The sum of two numbers is [latex]-45[/latex]. Their difference is [latex]-89[/latex]. Find the numbers.
46. Andrea is buying some new shirts and sweaters. She is able to buy [latex]3[/latex] shirts and [latex]2[/latex] sweaters for [latex]$114[/latex] or she is able to buy [latex]2[/latex] shirts and [latex]4[/latex] sweaters for [latex]$164[/latex]. How much does a shirt cost? How much does a sweater cost?
Solution

A shirt costs [latex]$16[/latex] and a sweater costs [latex]$33[/latex].

47. Peter is buying office supplies. He is able to buy [latex]3[/latex] packages of paper and [latex]4[/latex] staplers for [latex]$40[/latex] or he is able to buy [latex]5[/latex] packages of paper and [latex]6[/latex] staplers for [latex]$62[/latex]. How much does a package of paper cost? How much does a stapler cost?
48. The total amount of sodium in [latex]2[/latex] hot dogs and [latex]3[/latex] cups of cottage cheese is [latex]4720[/latex] mg. The total amount of sodium in [latex]5[/latex] hot dogs and [latex]2[/latex] cups of cottage cheese is [latex]6300[/latex] mg. How much sodium is in a hot dog? How much sodium is in a cup of cottage cheese?
Solution

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

49. The total number of calories in [latex]2[/latex] hot dogs and [latex]3[/latex] cups of cottage cheese is [latex]960[/latex] calories. The total number of calories in [latex]5[/latex] hot dogs and [latex]2[/latex] cups of cottage cheese is [latex]1190[/latex] 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.[latex]\left\{\begin{array}{c}8x-15y=-32\\ 6x+3y=-5\end{array}\right.[/latex]
b.[latex]\left\{\begin{array}{c}x=4y-3\\ 4x-2y=-6\end{array}\right.[/latex]

Solution

a. elimination
b. substitution

51.

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

52.

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

Solution

a. substitution
b. elimination

53.

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

Exercises: Everyday Math

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

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

a. for [latex]r[/latex], his rowing speed in still water.
b. Then solve for [latex]c[/latex], the speed of the river current.

 

Solution

a. [latex]r=4[/latex]
b. [latex]c=1[/latex]

55. Josie wants to make [latex]10[/latex] pounds of trail mix using nuts and raisins, and she wants the total cost of the trail mix to be [latex]$54[/latex]. Nuts cost [latex]$6[/latex] per pound and raisins cost [latex]$3[/latex] per pound. Solve the system [latex]\left\{\begin{array}{c}n+r=10\\ 6n+3r=54\end{array}\right.[/latex] to find [latex]n[/latex], the number of pounds of nuts, and [latex]r[/latex], 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 [latex]\left\{\begin{array}{c}x+y=10\\ 5x+8y=56\end{array}\right.[/latex]

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

Solution

a. (8, 2)
b.

This image is a graph that shows the solution to the system “x plus y equals 10” and 5x plus 8y equals 56. The solution is on an x, y coordinate plane. Two arrows intersect at points 8 and 2.
Figure 4P.3.1

c. Answers will vary.

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

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

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Fanshawe Pre-Health Sciences Mathematics 1 Copyright © 2022 by Domenic Spilotro, MSc is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License, except where otherwise noted.

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