Instructions: For questions 1-8, determine if the following points are solutions to the given system of equations.
1. [latex]\left\{\begin{array}{c}2x-6y=0\\3x-4y=5\end{array}\right.[/latex]
a.[latex](3,1)[/latex]
b.[latex](-3,4)[/latex]
2. [latex]\left\{\begin{array}{c}7x-4y=-1\\-3x-2y=1\end{array}\right.[/latex]
a. [latex](1,2)[/latex]
b. [latex](1,-2)[/latex]
3. [latex]\left\{\begin{array}{c}2x+y=5\\ x+y=1\end{array}\right.[/latex]
a.[latex](4,3)[/latex]
b.[latex](2,0)[/latex]
4. [latex]\left\{\begin{array}{c}-3x+y=8\\ -x+2y=-9\end{array}\right.[/latex]
a.[latex](-5,-7)[/latex]
b.[latex](-5,7)[/latex]
5. [latex]\left\{\begin{array}{c}x+y=2\\ y=\frac{3}{4}x\end{array}\right.[/latex]
a.[latex]\left(\frac{8}{7},\frac{6}{7}\right)[/latex]
b.[latex]\left(1,\frac{3}{4}\right)[/latex]
6. [latex]\left\{\begin{array}{c}x+y=1\\y=\frac{2}{5}x\end{array}\right.[/latex]
a.[latex]\left(\frac{5}{7},\frac{2}{7}\right)[/latex]
b.[latex](5,2)[/latex]
7. [latex]\left\{\begin{array}{c}x+5y=10\\ y=\frac{3}{5}x+1\end{array}\right.[/latex]
a.[latex](-10,4)[/latex]
b.[latex]\left(\frac{5}{4},\frac{7}{4}\right)[/latex]
8. [latex]\left\{\begin{array}{c}x+3y=9\\ y=\frac{2}{3}x-2\end{array}\right.[/latex]
a.[latex](-6,5)[/latex]
b.[latex]\left(5,\frac{4}{3}\right)[/latex]
Instructions: For questions 9-50, solve the following systems of equations by graphing.
9. [latex]\left\{\begin{array}{c}3x+y=-3\\ 2x+3y=5\end{array}\right.[/latex]
Solution
[latex](-2,3)[/latex]
10. [latex]\left\{\begin{array}{c}-x+y=2\\ 2x+y=-4\end{array}\right.[/latex]
11. [latex]\left\{\begin{array}{c}-3x+y=-1\\ 2x+y=4\end{array}\right.[/latex]
Solution
[latex](1,2)[/latex]
12. [latex]\left\{\begin{array}{c}-2x+3y=-3\\ x+y=4\end{array}\right.[/latex]
13. [latex]\left\{\begin{array}{c}y=x+2\\ y=-2x+2\end{array}\right.[/latex]
Solution
[latex](0,2)[/latex]
14. [latex]\left\{\begin{array}{c}y=x-2\\ y=-3x+2\end{array}\right.[/latex]
15. [latex]\left\{\begin{array}{c}y=\frac{3}{2}x+1\\ y=-\frac{1}{2}x+5\end{array}\right.[/latex]
Solution
[latex](2,4)[/latex]
16. [latex]\left\{\begin{array}{c}y=\frac{2}{3}x-2\\ y=-\frac{1}{3}x-5\end{array}\right.[/latex]
17. [latex]\left\{\begin{array}{c}-x+y=-3\\ 4x+4y=4\end{array}\right.[/latex]
Solution
[latex](2,-1)[/latex]
18. [latex]\left\{\begin{array}{c}x-y=3\\ 2x-y=4\end{array}\right.[/latex]
19. [latex]\left\{\begin{array}{c}-3x+y=-1\\ 2x+y=4\end{array}\right.[/latex]
Solution
[latex](1,2)[/latex]
20. [latex]\left\{\begin{array}{c}-3x+y=-2\\ 4x-2y=6\end{array}\right.[/latex]
21. [latex]\left\{\begin{array}{c}x+y=5\\ 2x-y=4\end{array}\right.[/latex]
Solution
[latex](3,2)[/latex]
22. [latex]\left\{\begin{array}{c}x-y=2\\ 2x-y=6\end{array}\right.[/latex]
23. [latex]\left\{\begin{array}{c}x+y=2\\ x-y=0\end{array}\right.[/latex]
Solution
[latex](1,1)[/latex]
24. [latex]\left\{\begin{array}{c}x+y=6\\ x-y=-8\end{array}\right.[/latex]
25. [latex]\left\{\begin{array}{c}x+y=-5\\ x-y=3\end{array}\right.[/latex]
Solution
[latex](-1,-4)[/latex]
26. [latex]\left\{\begin{array}{c}x+y=4\\ x-y=0\end{array}\right.[/latex]
27. [latex]\left\{\begin{array}{c}x+y=-4\\ -x+2y=-2\end{array}\right.[/latex]
Solution
[latex](3,3)[/latex]
28. [latex]\left\{\begin{array}{c}-x+3y=3\\ x+3y=3\end{array}\right.[/latex]
29. [latex]\left\{\begin{array}{c}-2x+3y=3\\ x+3y=12\end{array}\right.[/latex]
Solution
[latex](-5,6)[/latex]
30. [latex]\left\{\begin{array}{c}2x-y=4\\ 2x+3y=12\end{array}\right.[/latex]
31. [latex]\left\{\begin{array}{c}2x+3y=6\\ y=-2\end{array}\right.[/latex]
Solution
[latex](6,-2)[/latex]
32. [latex]\left\{\begin{array}{c}-2x+y=2\\ y=4\end{array}\right.[/latex]
33. [latex]\left\{\begin{array}{c}x-3y=-3\\ y=2\end{array}\right.[/latex]
Solution
[latex](3,2)[/latex]
34. [latex]\left\{\begin{array}{c}2x-2y=8\\ y=-3\end{array}\right.[/latex]
35. [latex]\left\{\begin{array}{c}2x-y=-1\\ x=1\end{array}\right.[/latex]
Solution
[latex](1,3)[/latex]
36. [latex]\left\{\begin{array}{c}x+2y=2\\ x=-2\end{array}\right.[/latex]
37. [latex]\left\{\begin{array}{c}x-3y=-6\\ x=-3\end{array}\right.[/latex]
Solution
[latex](-3,1)[/latex]
38. [latex]\left\{\begin{array}{c}x+y=4\\ x=1\end{array}\right.[/latex]
39. [latex]\left\{\begin{array}{c}4x-3y=8\\ 8x-6y=14\end{array}\right.[/latex]
40. [latex]\left\{\begin{array}{c}x+3y=4\\ -2x-6y=3\end{array}\right.[/latex]
41. [latex]\left\{\begin{array}{c}-2x+4y=4\\ y=\frac{1}{2}x\end{array}\right.[/latex]
42. [latex]\left\{\begin{array}{c}3x+5y=10\\ y=-\frac{3}{5}x+1\end{array}\right.[/latex]
43. [latex]\left\{\begin{array}{c}x=-3y+4\\ 2x+6y=8\end{array}\right.[/latex]
44. [latex]\left\{\begin{array}{c}4x=3y+7\\ 8x-6y=14\end{array}\right.[/latex]
45. [latex]\left\{\begin{array}{c}2x+y=6\\ -8x-4y=-24\end{array}\right.[/latex]
Solution
infinitely many solutions
46. [latex]\left\{\begin{array}{c}5x+2y=7\\ -10x-4y=-14\end{array}\right.[/latex]
47. [latex]\left\{\begin{array}{c}x+3y=-6\\ 4y=-\frac{4}{3}x-8\end{array}\right.[/latex]
Solution
infinitely many solutions
48. [latex]\left\{\begin{array}{c}-x+2y=-6\\ y=-\frac{1}{2}x-1\end{array}\right.[/latex]
49. [latex]\left\{\begin{array}{c}-3x+2y=-2\\ y=-x+4\end{array}\right.[/latex]
Solution
[latex]\left(2,2\right)[/latex]
50. [latex]\left\{\begin{array}{c}-x+2y=-2\\ y=-x-1\end{array}\right.[/latex]
Instructions: For questions 51-62, determine the number of solutions and then classify the system of equations without graphing.
51. [latex]\left\{\begin{array}{c}y=\frac{2}{3}x+1\\ -2x+3y=5\end{array}\right.[/latex]
Solution
[latex]0[/latex] solutions
52. [latex]\left\{\begin{array}{c}y=\frac{1}{3}x+2\\ x-3y=9\end{array}\right.[/latex]
53. [latex]\left\{\begin{array}{c}y=-2x+1\\ 4x+2y=8\end{array}\right.[/latex]
Solution
[latex]0[/latex] solutions
54. [latex]\left\{\begin{array}{c}y=3x+4\\ 9x-3y=18\end{array}\right.[/latex]
55. [latex]\left\{\begin{array}{c}y=\frac{2}{3}x+1\\ 2x-3y=7\end{array}\right.[/latex]
Solution
no solutions, inconsistent, independent
56. [latex]\left\{\begin{array}{c}3x+4y=12\\ y=-3x-1\end{array}\right.[/latex]
57. [latex]\left\{\begin{array}{c}4x+2y=10\\ 4x-2y=-6\end{array}\right.[/latex]
Solution
consistent, [latex]1[/latex] solution
58. [latex]\left\{\begin{array}{c}5x+3y=4\\ 2x-3y=5\end{array}\right.[/latex]
59. [latex]\left\{\begin{array}{c}y=-\frac{1}{2}x+5\\ x+2y=10\end{array}\right.[/latex]
Solution
infinitely many solutions
60. [latex]\left\{\begin{array}{c}y=x+1\\ -x+y=1\end{array}\right.[/latex]
61. [latex]\left\{\begin{array}{c}y=2x+3\\ 2x-y=-3\end{array}\right.[/latex]
Solution
infinitely many solutions
62. [latex]\left\{\begin{array}{c}5x-2y=10\\ y=\frac{5}{2}x-5\end{array}\right.[/latex]
Instructions: For questions 63-66, solve.
63. Molly is making strawberry infused water. For each ounce of strawberry juice, she uses three times as many ounces of water. How many ounces of strawberry juice and how many ounces of water does she need to make [latex]64[/latex] ounces of strawberry infused water?
Solution
Molly needs [latex]16[/latex] ounces of strawberry juice and [latex]48[/latex] ounces of water.
64. Jamal is making a snack mix that contains only pretzels and nuts. For every ounce of nuts, he will use [latex]2[/latex] ounces of pretzels. How many ounces of pretzels and how many ounces of nuts does he need to make [latex]45[/latex] ounces of snack mix?
65. Enrique is making a party mix that contains raisins and nuts. For each ounce of nuts, he uses twice the amount of raisins. How many ounces of nuts and how many ounces of raisins does he need to make [latex]24[/latex] ounces of party mix?
Solution
Enrique needs [latex]8[/latex] ounces of nuts and [latex]16[/latex] ounces of water.
66. Owen is making lemonade from concentrate. The number of quarts of water he needs is [latex]4[/latex] times the number of quarts of concentrate. How many quarts of water and how many quarts of concentrate does Owen need to make [latex]100[/latex] quarts of lemonade?
