PMI Systems of Equations - PreCalculus
Solve by Graphing
Class Work
1. y = -x – 7
4
y = 3x – 7
1
2. y = - 4x + 2
1
2
y=- x+3
3
4
3
x
4
6. y = x – 3
y=
4
3
+2
7. y = x + 3
2
3
y=- x–3
1
3. y = -3x – 5
y=x+3
8. y = 3x + 2
y = -x – 2
4. y = -2x + 5
1
y = 3x – 2
9. y = 4x – 1
y = -x + 4
5. y = -4x + 7
y = -3x + 3
10. y = 3x – 4
y = 4x + 10
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PMI Systems of Equations - PreCalculus
Solving by Graphing
Homework
3
11. y = - 2x – 4
y=- 𝑥+1
16. y = -4x – 1
y = x – 11
12. y = -2x – 2
y = -3x – 6
17. y = -3x – 3
1
y = 2x + 4
13. y = x – 2
y=x+2
18. y = - x + 3
1
2
3
14. y = 4x + 1
y=-
1
x
2
–4
15. y = x – 4
y = -x + 2
2
2
5
y = 5x – 1
19. y = -x – 2
1
y = - 2x + 2
20. y = x + 5
y = -x + 3
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PMI Systems of Equations - PreCalculus
Solve by Substitution
Class Work
21. x = 4y – 9
x=y+3
26. x = 5y – 38
x = -4y + 16
22. 5x = -2y + 48
x = -3y + 20
27. y = 2x + 3
4x – 2y = 8
23. y – 4x = 28
y = -2x – 2
28. x = -4y + 8
x = 3y + 8
24. y + 2x = -12
y = x + 15
29. 5y + 5x = 85
y = 4x – 18
25. x = -2y – 7
2x + y = -14
30. x = y – 12
x = 5y – 40
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PMI Systems of Equations - PreCalculus
Solving by Substitution
Homework
31. y = -5x + 41
-2x = -14 – 2y
36. y = -2x + 11
5y – 2x = 31
32. y = 3x + 6
-6x + 2y = 12
37. 5y – 5x = -15
y = -3x + 29
33. y – 3x = 0
y = -3x – 18
38. -4x = 3y + 32
x = -5y – 8
34. x = -3y + 13
4x – 4y = 20
39. y = -3x – 1
-4y + x = -9
35. x = -4y + 29
5x + 2y = 37
40. y = -4x + 17
-3y – x = -7
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PMI Systems of Equations - PreCalculus
Solve by Elimination (Addition & Subtraction)
Class Work
41. 3x + y = 36
5x + y = 56
46. -x + 2y = 14
x – 2y = -11
42. x + 2y = 25
x + 3y = 33
47. 4x – y = 16
4x + 2y = 16
43. 3x – 5y = -52
x – 5y = -34
48. 2x + 5y = 5
-2x + y = -23
44. 2x + 3y = 4
-2x + 5y = 60
49. 2x – 2y = -12
x – 2y = -13
45. 2x + 2y = 2
5x – 2y = 40
50. 5x + 5y = 40
-5x + 3y = -40
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PMI Systems of Equations - PreCalculus
Solve by Elimination (Addition & Subtraction)
Homework
51. 4x – y = -2
4x + 5y = 10
56. x + 5y = -4
-x + 2y = -10
52. 2x + 4y = 10
-4x + 4y = 52
57. -4x + 2y = -44
4x + 4y = 20
53. -3x – 5y = 49
3x + 4y = -44
58. x + 2y = 4
x + 5y = -2
54. -4x + 3y = 39
5x – 3y = -45
59. 3x – y = -5
-3x – 2y = -10
55. -5x – 2y = -5
-x – 2y = -1
60. 3x – y = 11
-3x – 5y = -71
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PMI Systems of Equations - PreCalculus
Solve by Elimination (Multiply First)
Class Work
61. 5x – 4y = 47
-x – 16y = 125
66. 2x + 5y = -7
8x + 3y = 57
62. 3x – 2y = 33
-4x – 4y = 16
67. 4x + 3y = 33
8x + y = 31
63. 2x + y = 21
4x + 3y = 51
68. 3x + 3y = 31
-9x – 5y = -67
64. -3x + 3y = -27
12x + 5y = 108
69. –x – y = -8
-4x + 2y = 22
65. 3x + 4y = 3
-12x – y = -57
70. 2x + y = 0
-8x + 4y = 80
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PMI Systems of Equations - PreCalculus
Solve by Elimination (Multiply First)
Homework
71. –x + y = -5
-3x + 4y = -12
76. x + 5y = -12
3x + y = 6
72. -2x – y = 2
-6x + 3y = -18
77. x + y = 14
4x – 2y = 2
73. -2x + 2y = 16
6x – y = -13
78. -3x + 3y = -3
-12x + 5y = -61
74. -4x – 5y = -9
3x + 10y = 13
79. 3x + 2y = 27
-9x + 4y = -51
75. 3x – 2y = -26
6x – 4y = -70
80. 4x + 4y = 20
2x – 16y = -44
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PMI Systems of Equations - PreCalculus
Choose Your Own Strategy
Class Work
81. –x + 4y = 5
x + 4y = 11
82. 3x – y = 7
4x – 2y = 8
83. 2y + 5x = 35
y = 4x – 28
84. 5x – 4y = -39
-3x – 4y = -15
85. y = -5x + 59
4x + y = 49
86. y = x + 6
4
y = 5x + 6
87. -2x + 4y = 28
2x – 3y = -18
88. y = -x + 12
3y + 3x = 36
89. 2x + 4y = -10
-4x – 12y = 36
90. –x – 5y = -3
-2x + 5y = 9
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PMI Systems of Equations - PreCalculus
Choose Your Own Strategy
Homework
91. -3x + 5y = -39
12x – 4y = 60
92. y = 3x – 18
y – 3x = -24
4
3
96. y = - x + 4
2
y = 3x + 10
97. 3x + 2y = 21
-3x + 5y = 21
93. x + 3y = 16
-x + 4y = 5
98. –x – y = 7
-x + 5y = 19
94. x = -3y – 19
x + 5y = -22
99. 4x + y = -28
-2x + 2y = 24
95. 3x – 3y = 12
9x + 2y = 102
100.
x = 2y – 7
-x + 4y = 17
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PMI Systems of Equations - PreCalculus
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Solving Systems of Non-linear Equations
Class Work
101. y = x2 + 2x – 3
y = -x – 4
106.
y = x3 – 2
y = 2x + 3
102.
y = x2 – 2x – 1
y = -x2 – 4x + 3
107.
y = x3 - 2
y = x3 + 2
103.
y = (x – 4)(x + 2)
y = -(x + 8)(x + 1)
108.
y = 3x2 + 9x
y = (x + 3)3
104.
y = x2
109.
𝑦 = − 2 𝑥2 − 3
1
𝑦 = 3 𝑥2 − 3
y = x3
105. y = x2 – 2x – 3
𝑦=
1
− 3 𝑥3
1
−2
110.
y = x3 + 3x2 – 6x - 12
y = 12x + 28
PMI Systems of Equations - PreCalculus
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Solving Systems of Non-linear Equations
Homework
111. y = (x + 2)3 + 1
y=x+3
116.
y = -x2 – 4x – 1
y=x+3
112.
y = x2 + 8x + 19
y = -x2 – 8x – 11
117.
y = -x2 – 4x +7
y = -x3 + 3
113.
y = -x
y = -x2 + 4
118.
y = -(x + 2)2 + 4
y = -x2
114.
y = x2 – 3x + 4
119.
𝑦 = 2 𝑥2 − 3
1
𝑦 = − 3 𝑥2 − 2
y = -x2 + 2x + 3
115.
y = x2 + x – 6
y = (x + 3)(x – 2)
1
120.
1
𝑦 = − 2 𝑥2 + 𝑥 + 4
y = x2 + 2x + 1
PMI Systems of Equations - PreCalculus
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Writing Systems to Model Situations
Class Work
Solve each problem by writing a system of equations and using it to solve the problem.
Write your answers in complete sentences.
121. The admission fee at a carnival is $3.00 for children and $5.00 for adults. On the first
day 1,500 people enter the fair and $5740 is collected. How many children and how many
adults attended the carnival?
122. A builder placed two orders with the hardware store. The first order was for 25 sheets of
plywood and 4 boxes of nails and the bill totaled $357. The second order was for 35 sheets
of plywood and 2 boxes of nails and the bill totaled $471. The bills do not list the per-item
price. What were the prices of one piece of plywood and one box of nails?
123. The student council is running a contest to guess the number of dimes in a jar that has a
combination of dimes and nickels. The total value in the jar is $12 and there are 15 more
nickels than dimes. How many dimes are there?
124. In your math class 70% of your grade is based on your test average and 30% on your
quiz average. Your test average is 10 points higher than your quiz average. If your quiz
average is 87%, what is your test average?
PMI Systems of Equations - PreCalculus
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Writing Systems to Model Situations
Homework
Solve each problem by writing a system of equations and using it to solve the problem.
Write your answers in complete sentences.
125. Two friends bought some markers and pens. The first bought 4 markers and 5 pens and
it cost him $6.71. The second friend bought 5 markers and 3 pens, which cost her $7.12.
What is the price for one marker and one pen?
126. The ticket price for the movies is $7.50 for children and $10.50 for adults. One night 825
people bought tickets and $8005.50 was collected from ticket sales. How many children and
how many adults bought tickets.
127. A plane flying from Newark to Los Angeles flew into a head wind, resulting in the 3000
mile trip taking 7 and a half hours. On the return trip, the plane and wind speed remained
the same, but because the plane flew with the wind the trip took only 5 hours. What was the
planes speed? (d = rt)
128. Ted is 5 years older than Cal. In three years, Cal will be 4/5 of Ted’s age. How old were
they 2 years ago?
PMI Systems of Equations - PreCalculus
Solving Systems of Inequalities by Graphing
Class Work
129.
5
y ≤ 2x – 2
133.
y > 2x – 5
3x + 4y < 12
134.
x+y≥ -3
5x – y ≤ -3
1
y ≥ 2x + 2
1
130.
y ≤ 4x – 1
y < 5x – 5
131.
y > - 3x – 3
1
y≤
132.
5
x
3
+3
y > x2 – 2x – 1
y ≤ -2x2 + 4
135. y >x3 – 1
y < -x2 + 4
136. y ≥ -x3 – 1
y ≤ -x – 1
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PMI Systems of Equations - PreCalculus
Solving Systems of Inequalities by Graphing
Homework
1
137.
y < 2x + 2
y > 3x – 3
141.
y > 2x + 4
2x – y ≤ 4
138.
y ≥ -x + 5
y ≤ 3x – 4
142.
3x – y ≥ - 1
x + y ≤ -3
139.
y < 4x
y ≤ -x + 4
1
140. y > -x2 + 5
y<4
143. y ≥ x2 – 2x – 2
y ≤ -x2 + 3
144. y >(x – 3)3 – 2
y ≥ -x3 – 2
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PMI Systems of Equations - PreCalculus
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Linear Programming
Class Work
Graph each system. Use the solution to maximize and minimize the profit function.
145.
x>0
y>0
x+y<4
P(x,y) = 3x – 2y
148.
146.
149.
x>0
y>0
y < -2x + 6
P(x,y) = x+y
147.
y>0
y < -x + 4
y > -2x + 4
P(x,y) = 3x +2xy
y < -x + 4
y > -2x +4
y>2
P(x,y) = 2xy
0<x<4
2<y<5
P(x,y) = 2xy – y
150. The new store “kitchens and baths” sells furniture for kitchens and baths. Each kitchen
display is 300 sq ft and each bath is 200 sq ft. The store has 6000 sq ft of display space. If
kitchens sell for $4000 and baths for $2500, how many of each should the store display to
maximize profit potential?
PMI Systems of Equations - PreCalculus
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Linear Programming
Homework
Graph each system. Use the solution to maximize and minimize the profit function.
151.
x>0
y>0
x+y<6
P(x,y) = 3x – 2y
154.
152.
155.
x>0
y>0
y < -3x + 6
P(x,y) = x+y
x>0
y < -1x+4
y>x
P(x,y) = 3y +2xy
x>0
y < -x + 6
y > -2x +4
y>2
P(x,y) = 2y - x
153.
0<x<5
6<y<8
P(x,y) = 2xy – y
156. The new store “kitchens and baths” sells furniture for kitchens and baths. Each kitchen
display is 400 sq ft and each bath is 300 sq ft. The store has 6000 sq ft of display space. If
kitchens sell for $5000 and baths for $4000, how many of each should the store display to
maximize profit potential?
PMI Systems of Equations - PreCalculus
Systems of Equations with 3-Variables
Class Work
Solve the following systems.
157.
2x + y + z = 9
x – y + 2z = 9
3x + y – 2z = -1
160.
x + 2y + 3z = 6
3x – y + z = 3
2x + y – 2z = 1
158.
x+y+z=2
2x – y + 3z = 13
x + y – z = -4
161.
2x – y + z = 5
x+y–z=1
2x + 3y + z = 13
159.
x + 4y + z = 4
2x – y – z = 5
x + 2y – z = 2
162.
x+y=0
y–z=3
x+z=-3
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PMI Systems of Equations - PreCalculus
Systems of Equations with 3-Variables
Homework
Solve the following systems
163. 2x +y – z = 2
x+y+z=6
x + 2y + 3z = 13
166.
x + y + z =7
2x – y +3z =13
x + 4y – 2z = 4
164.
2x – y + z = 3
x + 2y + 4z = 5
2x – y – 3z = - 1
167.
x + 2y + z = -8
2x – 2y – z = 5
x + 3y + 2z = -13
165.
3x + y + 2z = 5
2x + 2y – z = 2
x + 2y + z = 0
168.
x+y=3
z–x=2
y – z = -3
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PMI Systems of Equations - PreCalculus
Multiple Choice
1. The solution to y = 3x -1 and y = x + 7 is
a. (-1, 7)
b. (3, 1)
c. (2, 5)
d. (4, 11)
2. (1, 3) is the solution to
a. y = 3x and y = 7x – 4
b. y = 2x +1 and y = 5x – 1
c. y = 4x – 1 and y = 3x + 1
d. y = x + 2 and y = x + 3
3. If the graph of a system shows to parallel lines with unique y-intercepts
a. there is infinite solutions
b. there is no solution
c. there may be a solution if the lines are extended
d. something is incorrect with the graph
4. Bob wants to sub 2x + y = 3 into 3x + 3y = 6, he needs to
a. convert the equation to 2x = 3 – y
b. convert the equation y = 2x + 3
c. convert the equation to y = 3 – 2x
d. convert the equation to x = 1.5 – y
5. To solve 2x + 3y = 7 and x + y = 3 by elimination
a. first multiply the second equation by 2
b. first multiply the second equation by -3
c. first multiply the second equation by -6
d. first multiply the second equation by 6
6. Solve x + y = 7 and x – y = 3
a. (1, 6)
b. (2, 5)
c. (5, 2)
d. no solution
7. Solve 3x + 2y = 8 and 2x + 3y = 7
a. (2, 1)
b. (1, 3)
c. (3, -.5)
d. no solution
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PMI Systems of Equations - PreCalculus
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8. Solve 2x + y = 8 and 6x + 3y =10
a. (4, 0)
b. (3, 2)
c. (5, -2)
d. no solution
9. A motor boat traveling with the current travels at a rate of 7 miles per hour. Traveling
against the current the boat goes 3 miles per hour. How fast is the current?
a. 2 mph
b. 3 mph
c. 4 mph
d. 5 mph
10. The solution to y < 2x + 1 and y < 2x + 4 is
a. y < 2x + 1
b. y < 2x + 4
c. no solution
d. the shade region between the two boundaries
11. Which point in the system of y > 0, x > 0 and x + y < 7 will maximize P(x,y) = 2xy – 3y?
a. (0, 0)
b. (0, 7)
c. (7, 0)
d. (6, 1)
12. Which point in the system of y > -2x + 7, x + y < 7, and y > 1 will minimize P(x,y) = x + y?
a. (0, 0)
b. (0, 7)
c. (6, 1)
d. (3, 1)
13. The solution to the system x + y – z = 4, x + 2y + z = 11, and 2x –y + z = 5 is
a. (4, 1, 1)
b. (1, 2, -1)
c. (3, 3, 2)
d. (7, 0, 3)
14. The solution to the system x – y = 7, y + z = 1, and x – z = 2 is
a. (5, -2, 3)
b. (6, -1, 2)
c. (4, -3, 4)
d. (7, 0, 1)
PMI Systems of Equations - PreCalculus
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Extended Response
1. Given the equation y = 4x +2
a. write an equation for a line such it intersects the given line only at (0, 2)
b. write an equation for a line such that it never intersects the given line.
c. write an equation of a line such that it has infinite intersection with the given line.
2. For the school play there are three types of tickets: Adult, children and student. The adult
ticket is $10, the student ticket is $5, and a children’s ticket is $3. $730 worth of tickets were
sold and a total 90 people saw the show. If twice as many students as children saw the show
how many adults saw the show.
3. Given the system ax + by = c and 6x + 10y = 8
a. find a, b, and c such that the system has no solution
b. find a, b, and c such that the system has infinite solutions
c. find a, b, and c such that the system has the solution (-2, 2)
4. Create a system of inequalities that has a solution bound by the x-axis, the y-axis, and a line
with slope -3 passes through (1,3).
 what point of the system you created maximizes P(x,y) = 2x - 3y
 show that a point on the interior of the feasible region won’t be a max or a min.