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