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Advanced Algebra II
Ch. 3 Homework Packet
Section 3.1 – Solving Systems Using Tables and Graphs
Solve each system by graphing or using a table. Check your answers.
𝑦 =𝑥−2
𝑥 − 2𝑦 = 10
1.{
2. {
𝑥 + 𝑦 = 10
𝑦 = 𝑥 − 11
3. {
𝑥 + 𝑦 = −1
𝑥−𝑦 =3
4. {
2
𝑦 = 3𝑥 − 5
5. {
2
𝑦 = −3𝑥 − 3
6. {
4𝑥 + 3𝑦 = −16
−𝑥 + 𝑦 = 4
2𝑥 − 4𝑦 = −4
3𝑥 − 𝑦 = 4
Write and solve a system of equations for each situation. Check your answers.
7. Your school sells tickets for its winter concert. Student tickets are $5 and adult tickets are $10. If your
school sells 85 tickets and makes $600, how many of each ticket did they sell?
8. The spreadsheet below shows the monthly income and expenses for a new business.
A. Use your graphing calculator to find linear models for income and expenses as functions of the
number of the month.
B. In what month will income equal expenses?
Without graphing, classify each system as independent, dependent, or inconsistent.
𝑥+𝑦 =3
𝑥 + 3𝑦 = 9
𝑥+𝑦 =4
9. {
10. {
11. {
𝑦 = 2𝑥 − 3
−2𝑥 − 6𝑦 = −18
𝑦 = 2𝑥 + 1
12. {
3𝑥 + 2𝑦 = 7
3𝑥 − 15 = −6𝑦
13. {
𝑥 + 𝑦 = 11
𝑥 − 4𝑦 = −6
14. {
25𝑥 − 10𝑦 = 0
2𝑦 = 5𝑥
15. You are going on vacation and leaving your dog in a kennel. Kennel A charges $25 per day which
includes a one-time grooming treatment. Kennel B charges $20 per day and a one-time fee of $30 for
grooming.
a. Write a system of equations to represent the cost c for d days that your dog will stay at the kennel.
b. If your vacation is 7 days long, which kennel should you choose? Explain.
16. Write a second equation for the system so that the system will have the indicated number of
solutions.
No solutions
𝑦 = −𝑥 + 3
{
?
17. Which ordered pair of numbers is the solution of the system?
{
𝑥 + 𝑦 = −3
𝑥 − 2𝑦 = 0
A. (-6, -3)
B. (-2, -1)
C. (6, -3)
D. (2, 1)
Section 3.2 – Solving Systems Algebraically
Solve each system by substitution. Check your answers.
𝑦 =𝑥+1
5𝑥 − 𝑦 = −3
18. {
19. {
2𝑥 + 𝑦 = 7
𝑦 = 2𝑥 + 3
20. {
𝑥 + 3𝑦 = −4
𝑦+𝑥 =0
21. {
3𝑥 + 2𝑦 = 9
𝑥+𝑦 =3
22. Suppose you bought eight oranges and one grapefruit for a total of $4.60. Later that day, you bought
six oranges and three grapefruits for a total of $4.80. What is the price of each type of fruit?
Solve each system by elimination.
𝑥 + 𝑦 = 10
23. {
𝑥−𝑦 =2
25. {
𝑥 + 2𝑦 = 10
3𝑥 − 𝑦 = 9
24. {
𝑥+𝑦 =7
𝑥 + 3𝑦 = 11
26. {
𝑥−𝑦 =0
𝑥+𝑦 =2
27. {
3𝑥 − 𝑦 = 17
2𝑥 + 𝑦 = 8
28. {
5𝑥 + 4𝑦 = 2
−5𝑥 − 2𝑦 = 4
29. {
14𝑥 + 2𝑦 = 10
𝑥 − 5𝑦 = 11
30. {
0.3𝑥 + 0.4𝑦 = 0.8
0.7𝑥 − 0.8𝑦 = −6.8
31. You can buy DVDs at a local store for $15.49 each. You can buy them at an online store for $13.99
each plus $6 for shipping. How many DVDs can you buy for the same amount at the two stores?
32. Last year, a baseball team paid $20 per bat and $12 per glove, spending a total of $1040. This year,
the prices went up to $25 per bat and $16 per glove. The team spent $1350 to purchase the same amount
of equipment as last year. How many bats and gloves did the team purchase each year?
33. If the perimeter of the square is 72 units, what are the values of x and y?
Section 3.3 – Systems of Inequalities
Find all whole number solution of each system using a table.
𝑥
𝑦 < −3 +3
−𝑥 + 𝑦 = 1
34. {
35. {
𝑥 + 2𝑦 ≤ 20
2𝑥 + 𝑦 ≥ 4
36. The dry cleaner charges $4 to clean a pair of pants and $3 to clean a shirt. You want to get at least 8
items cleaned. You have $32 to spend on dry cleaning.
a. Write a system of inequalities to model the situation.
b. Solve the system by using a table.
Solve each system of inequalities by graphing.
𝑦 >𝑥+2
37. {
𝑦 ≤ −𝑥 + 1
38. {
𝑥+𝑦 <5
𝑦 < 3𝑥 − 2
39. {
2𝑥 ≥ 𝑦 + 3
𝑥 < 3 − 2𝑦
40. {
2𝑥 + 𝑦 > 2
𝑥−𝑦 ≥3
41. Suppose you are buying two kinds of notebooks for school. A spiral notebook costs $2, and a threering notebook costs $5. You must have at least 6 notebooks. The cost of the notebooks can be no more
than $20.
a. Write a system of inequalities to model the situation.
b. Graph and solve the system.
Solve each system of inequalities by graphing.
𝑦 <𝑥−3
42. {
𝑦 ≥ |𝑥 − 4|
43. {
−2𝑥 + 𝑦 > 1
𝑦 > |𝑥|
44. A doctor needs at least 60 adults for a medical study. He cannot use more than 40 men in the study.
Write a system of inequalities to model the situation and solve the system by graphing.
Section 3.4 – Linear Programming
Find the values of x and y that maximize or minimize the objective function for each graph.
45. Maximum for P = 6x + 2y
46. Maximum for P = x + y
47. Maximum for P = 2x + y
48. Minimum for P = 5x + 10y
Graph each system of constraints. Name all vertices. Then find the values of x and y that maximize or
minimize the objective function.
49. {
𝑥 + 2𝑦 ≤ 6
𝑥≥2
𝑦≥1
Minimum for C = 3x + 4y
𝑥+𝑦 ≤6
50. {2𝑥 + 𝑦 ≤ 10
𝑥 ≥ 0, 𝑦 ≥ 0
Maximum for P = 4x + y
51. Suppose you make and sell skin lotion. A quart of regular skin lotion contains 2 c oil and 1 c cocoa
butter. A quart of extra-rich skin lotion contains 1 c oil and 2 c cocoa butter. You will make a profit of
$10/qt on regular lotion and a profit of $8/qt on extra-rich lotion. You have 24 c oil and 18 c cocoa butter.
a. How many quarts of each type of lotion should you make to maximize your profit?
b. What is the maximum profit?
Section 3.5 – Systems With Three Variables
Solve each system by elimination. Check your answers.
𝑥 + 𝑦 + 𝑧 = −1
52. {2𝑥 − 𝑦 + 2𝑧 = −5
−𝑥 + 2𝑦 − 𝑧 = 4
𝑥 − 𝑦 + 2𝑧 = 10
53. { −𝑥 + 𝑦 − 2𝑧 = 5
3𝑥 − 3𝑦 + 6𝑧 = −2
𝑥 + 5𝑦 + 5𝑧 = −10
54. { 𝑥 + 𝑦 + 𝑧 = 2
𝑥 + 2𝑦 + 3𝑧 = −3
𝑥 + 𝑦 + 𝑧 = −2
55. {2𝑥 + 2𝑦 − 3𝑧 = 11
3𝑥 − 𝑦 + 𝑧 = 4
Solve each system by substitution. Check your answers.
𝑥 + 𝑦 + 4𝑧 = 5
56. { −2𝑥 + 2𝑧 = 3
3𝑥 + 𝑦 − 2𝑧 = 0
𝑥 + 𝑦 + 𝑧 = −8
57. { 𝑥 − 𝑦 − 𝑧 = 6
2𝑥 − 3𝑦 + 2𝑧 = −1
𝑥+𝑦+𝑧 =6
58. { 2𝑥 − 𝑦 + 2𝑧 = 6
−𝑥 + 𝑦 + 3𝑧 = 10
𝑥+𝑦−𝑧 =1
59. { 𝑥 + 2𝑧 = 3
2𝑥 + 2𝑦 = 4
60. Write and solve a system of equations. The sum of three numbers is -2. The sum of three times the
first number, twice the second number, and the third number is 9. The difference between the second
number and half the third number is 10. Find the numbers.
61. How do you decide whether substitution is the best method to solve a system in three variables?
62. The first number plus the third number is equal to the second number. The sum of the first number
and the second number is six more than the third number. Three times the first number minus two times
the second number is equal to the third number. What is the sum of the three numbers?
Section 3.6 – Solving Systems Using Matrices
Identify the indicated element.
63. A13
64. B24
65. B12
66. A22
67. B31
68. A21
69. B23
70. A11
71. Write a matrix to represent the system.
4𝑥 − 𝑦 + 2𝑧 = 10
{5𝑥 + 2𝑦 − 3𝑧 = 0
𝑥 − 3𝑦 + 𝑧 = 6
72. Write a matrix given the system of equations.
3𝑥 − 4𝑦 + 𝑧 = 15
{−2𝑥 − 6𝑦 + 3𝑧 = 4
2𝑥 + 2𝑦 − 2𝑧 = −1
73. Solve the system using a graphing calculator.
5𝑥 − 2𝑦 + 𝑧 = −1
{ −𝑥 − 𝑦 − 2𝑧 = 5
3𝑥 + 2𝑦 + 2𝑧 = 2
74. Suppose the movie theater you work at sells popcorn in three different sizes. A small costs $2, a
medium costs $5, and a large costs $10. On your shift, you sold 250 total containers of popcorn and
brought in $1726. You sold twice as many large containers as small ones.
a. How many of each popcorn size did you sell?
b. How much money did you bring in from selling small size containers?
75. The following matrix shows the prices passengers on an airline flight paid for a recent ticket and how
many passengers were on that flight. Some passengers paid full price for their tickets, and some bought
their tickets during a half-price sale. How many passengers bought each price of ticket?