Chapter 7 – Solving Systems of Equations and Inequalities
7.1
Answer Key
Graphs of Linear Systems
Answers 1. A system is a set of equations or inequalities with the same variables. OR A system is a set of algebraic sentences joined by the word “and”.
2. The solution to a system is the set of ordered pairs that are in common to each algebraic sentence. 3. To solve a system by graphing: 1) graph each equation in the system on the same set of axes 2) find the point(s) at which the graphs intersect 3) the point(s) of intersection are the solution to the system 4. One problem with using a graph to solve a system is that it requires very careful graphing, and even when that is done, it is likely that the graph can only give you an approximate value of the intersection (solution). If the intersection values are both integers, then it works, but in most cases another method would be necessary to find the exact solution. 5. The two main ways to write the solution to a system of equations are as an ordered pair (x, y) or as equivalent values: x=? and y=? 6. The graphs of each equation will intersect at (4, -6). Each graph will cross through the point (4, 6).
7. The “Intersection” command is located in the “Calculate” menu of your calculator. When in the graphing mode, if you press “2nd” then “Trace” you will see a list that includes “Intersection”. If you select this command the calculator will show you at what x and y values the two graphs you have entered cross or intersect.
8. Peter will be further from the starting line at 19.99 seconds and Nadia will be further from the starting line at 20.02 seconds. Peter has a 20 second head start and Nadia catches up with him at 20 seconds. At any time before 20 seconds, Peter will be ahead. At any time after 20 seconds Nadia will be ahead.
1
1
9. (2 , − 2) 10. (8, 13)
CK-12 Basic Algebra Concepts
1
Chapter 7 – Solving Systems of Equations and Inequalities
Answer Key
11. (−6, 20) 3 2
12. (3, − )
13.
14.
15.
CK-12 Basic Algebra Concepts
2
Chapter 7 – Solving Systems of Equations and Inequalities
Answer Key
16.
17.
18.
CK-12 Basic Algebra Concepts
3
Chapter 7 – Solving Systems of Equations and Inequalities
Answer Key
19.
20.
21.
CK-12 Basic Algebra Concepts
4
Chapter 7 – Solving Systems of Equations and Inequalities
Answer Key
22.
23. 6.6 years 1 3
24. 6 months (technically 5 )
25. 4 seconds (at 22 ft mark)
CK-12 Basic Algebra Concepts
5
Chapter 7 – Solving Systems of Equations and Inequalities
7.2
Answer Key
Systems Using Substitution
Answers 1. Example answer: When you have a system where at least one of the expressions is solved for 𝑦 or for 𝑥, the algebraic expression can be substituted for that variable in the other expressions in the system. You then solve for the remaining variable. Once you have a numeric value for that variable you can substitute it into any of the expressions in the system to find the numeric value of the second variable. 2. Example answer: Systems are easier to solve using substitution if one of the expressions already has one of the variables isolated on one side of the equation or if it would be easy to isolate one variable on one side of the equation.
3. (−1, −3)
1
4. (2 3 , −3)
5. (−4, 4)
6. (−1, 3)
7. (−1, 7)
8. (−7, −1)
9. (3, −2)
10. (5, 0)
11. Undefined
12. (−5, 7)
CK-12 Basic Algebra Concepts
6
Chapter 7 – Solving Systems of Equations and Inequalities
Answer Key
13. (7, −1) 14. (−9.88, −16.47) : Answers for Which do you prefer? Why? will vary.
15. 30°, 60°
16. 29.5, 40.5
17. 120 yards by 80 yards
18. The angles are 99° and 81°
CK-12 Basic Algebra Concepts
7
Chapter 7 – Solving Systems of Equations and Inequalities
7.3
Answer Key
Mixture Problems
Answers 1. 1.6lbs peanuts and 3.4lbs cashews 2. 0.8L of 10% solution and 0.2L of 35% solution
3. The bracelet density is 16.37 g/cc
4. Jeffrey needs 0.4 pounds of blueberries and 5.6 pounds of raspberries
5. 30 L of 5% and 70 L of 15%
CK-12 Basic Algebra Concepts
8
Chapter 7 – Solving Systems of Equations and Inequalities
7.4
Answer Key
Linear Systems with Addition or Subtraction
Answers
1. The purpose of the elimination method is to eliminate a variable in the system to make solving certain systems faster and easier. The method is appropriate when both equations are in standard form and graphing or using substitution would require a great deal of work. 2. (−5, −7)
3. (−1, −2)
4. (0, 2)
5. (5, 5)
6. (6, 2)
7. (−4, −2)
8. (3.5, −2)
9. (−2, −3)
10. (7.5, −1)
11. Candy bars cost $0.48 and fruit roll-ups cost $0.35
12. The wind speed is 24 miles per hour.
CK-12 Basic Algebra Concepts
9
Chapter 7 – Solving Systems of Equations and Inequalities
Answer Key
13. a. The pick-up fee is $0.85 b. The rate per mile is $1.12 c. A seven-mile trip is $8.69 14. $0.75 per minute for the first five minutes, and $0.25 per minute thereafter. 15. The plumber worked 6.25 hours and the builder worked 4 hours.
16. Paul makes $6.20 per hour with a $12 bonus per warranty.
CK-12 Basic Algebra Concepts
10
Chapter 7 – Solving Systems of Equations and Inequalities
7.5
Answer Key
Linear Systems with Multiplication
Answers
1.
35
2.
66
3.
90
4.
84
5.
34
6.
6
7.
6
8.
111
9.
126
10. 5
11. -2
12. -7
13. 8
CK-12 Basic Algebra Concepts
11
Chapter 7 – Solving Systems of Equations and Inequalities
Answer Key
14. 9 15. -11
16. -4
17. 0, -1.5
18. 3, 5
19. 1.25, 1.25
20. .667, .6
21. -1.143, -1.857
22. .065, .848
23. -9, -3
24. 1, 5
25. 4, 3
26. -2, 3
27. 1.625, .469
CK-12 Basic Algebra Concepts
12
Chapter 7 – Solving Systems of Equations and Inequalities
Answer Key
28. -.52, -.34 29. A = 114, B = 66
30. Lengths equal 51 and 99
31. $68,750 was invested in Company A, and $31,250 was invested in Company B
32. Sold 43 decorated cakes and 74 undecorated cakes.
33. John is 32 and Claire is 28
CK-12 Basic Algebra Concepts
13
Chapter 7 – Solving Systems of Equations and Inequalities
7.6
Answer Key
Consistent and Inconsistent Linear Systems
Answers
1. A system with no solutions is called an inconsistent system. 2. A consistent system has at least one solution (one place where the lines intersect)
3. It is a Coincident System – Lines with the same slope and y- intercept, therefore the lines completely overlap.
4. One point of intersection indicates that the lines are formed from linear equations.
5. The result of solving the system will result in an untrue solution, thus indicating no points of intersection, so our conclusion would be that the lines are parallel.
6. Consistent system with one solution.
7. -1, -4
8. Coincident System: Infinite solutions.
9. 1.875, 5.25
10. 1.227, .091
11. Coincident System: Infinite solutions.
12. -1, -4
CK-12 Basic Algebra Concepts
14
Chapter 7 – Solving Systems of Equations and Inequalities
Answer Key
13. Coincident System: Infinite solutions. 14. No solution - Parallel Lines
15. -2, -3
16. -4, 8
17. -1, -4
18. 1.44, -2.17
19. No Solution – Parallel Lines
20. .58, -46
21. Coincident System: Infinite solutions.
22. 1.8, .6
23. 0, 24
24. No Solution – Parallel Lines
25. Coincident System: Infinite solutions.
26. 30
CK-12 Basic Algebra Concepts
15
Chapter 7 – Solving Systems of Equations and Inequalities
Answer Key
27. 850 adults, 350 children 28. $23.00 per tie, and $47.00 per pair of suspenders
29. 600 mph in the Jetstream, and 480 mph in still air.
30. No real answers
CK-12 Basic Algebra Concepts
16
Chapter 7 – Solving Systems of Equations and Inequalities
7.7
Answer Key
Linear Programming
Answers
1. Linear Programming is the mathematical process of analyzing a system of inequalities to make the best decisions given the constraints of the situation. 2. When two linear equations are graphed on the same graph, the region where overlap occurs is the “feasible region”
3. Constraints are restrictions placed on a situation due to time, money or materials.
4. Once identified the feasible region is used to answer profitability or optimization questions.
5. Maximum or minimum values occur at the vertices of the feasible region (the points where the boundary lines intersect)
6.
CK-12 Basic Algebra Concepts
17
Chapter 7 – Solving Systems of Equations and Inequalities
Answer Key
7.
8.
9.
CK-12 Basic Algebra Concepts
18
Chapter 7 – Solving Systems of Equations and Inequalities
Answer Key
10.
11.
12.
CK-12 Basic Algebra Concepts
19
Chapter 7 – Solving Systems of Equations and Inequalities
Answer Key
13.
14. 𝑥 − 𝑦 < −6 𝑎𝑛𝑑 2𝑦 ≥ 3𝑥 + 17 15. 𝑥 + 4𝑦 ≤ 6 , 2𝑥 − 3𝑦 ≤ 21, 𝑎𝑛𝑑 3𝑥 + 4𝑦 ≥ −4 16. 2𝑥 + 𝑦 ≥ 12, 12𝑥 − 7𝑦 > 120, 𝑎𝑛𝑑 2𝑥 − 8𝑦 ≥ 36 17. Maximum value of z is 0, minimum is -20 18. Maximum value of z is 12, minimum is -17 19. 8 bookcases and 10 T.V. stands for a maximum profit of $1,480 20. $6750 in municipal bonds, $2250 in CD's, and $1000 in high risk
CK-12 Basic Algebra Concepts
20
Chapter 7 – Solving Systems of Equations and Inequalities
7.8
Answer Key
Probability and Permutations
Answers
1. A permutation is an arrangement of objects in a specific order. It is the product of the counting numbers 1 through n. 2. 5040
3. 3,628,800
4. 1
5. 120
6. 362,880
7. 6
8. 48
9. 2.1 𝑥 1013
10. 9506
11. 19958400
12. 301
CK-12 Basic Algebra Concepts
21
Chapter 7 – Solving Systems of Equations and Inequalities
Answer Key
13. 6720 14. 362882
15. 110
16. 120
17. 60
18. 10897286400
19. 8.3 𝑥 1081
20. 87178291200 𝑤𝑎𝑦𝑠
21. 5040
22. 1.3 𝑥 1015
23. 40320 24. a. 132 𝑤𝑎𝑦𝑠 b. 479001600
25. 720
26. .25 chance 27. 120
CK-12 Basic Algebra Concepts
22
Chapter 7 – Solving Systems of Equations and Inequalities
Answer Key
28. 720 29. a. 6 possibilities b. .33 of a chance
30. 1 out of 2652 chance. 31. a. 36 ways b. 45 ways
CK-12 Basic Algebra Concepts
23
Chapter 7 – Solving Systems of Equations and Inequalities
7.9
Answer Key
Probability and Combinations
Answers
1.
A combination is an arrangement of objects in no particular order. In a permutation, order matters.
2.
Number of items (n) divided by number of possible outcomes (r).
3.
Because n designates total number of items, k indicates how many of those items are chosen at one time. You cannot choose 9 items from a group of 3.
4.
66
5.
56
6.
5
7.
1
8.
1
9.
126
10. 184756
11. 19
12. 38760
CK-12 Basic Algebra Concepts
24
Chapter 7 – Solving Systems of Equations and Inequalities
Answer Key
13. 715 14. 35
15. 462
16. 5
17. 715
18. 15504
19. 1
20. 15
21. 48
22. 210
23. 378
24. 1935360
25. 1 out of 6
26. 1 out of 6
CK-12 Basic Algebra Concepts
25
Chapter 7 – Solving Systems of Equations and Inequalities
Answer Key
27. 4950 28. 35
29. 1365
30. a. 21 b. 12 c. 12
CK-12 Basic Algebra Concepts
26
