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# Chapter 7: Systems of Equations and Inequalities

Difficulty Level: Basic Created by: CK-12

## Introduction

James is trying to expand his pastry business to include cupcakes and personal cakes. He has a limited amount of manpower to decorate the new items and a limited amount of material to make the new cakes. In this chapter, you will learn how to solve this type of situation.

Every equation and inequality you have studied thus far is an example of a system. A system is a set of equations or inequalities with the same variables. This chapter focuses on the methods used to solve a system, such as graphing, substitution, and elimination. You will combine your knowledge of graphing inequalities to solve a system of inequalities.

Chapter Outline

## Summary

This chapter discusses linear systems of equations and the methods used to solve them, including graphing, substitution, addition, subtraction, and multiplication. Mixture problems are covered in detail, and a distinction is made between consistent and inconsistent linear systems. In addition, instruction is given on linear inequalities and linear programming, and the chapter concludes by talking about probability, permutations, and combinations.

### Systems of Equations and Inequalities; Counting Methods Review

1. Match the following terms to their definitions.
1. System – Restrictions imposed by time, materials, or money
2. Feasible Region – A system with an infinite number of solutions
3. Inconsistent System – An arrangement of objects when order matters
4. Constraints – A method used by businesses to determine the most profit or least cost given constraints
5. Consistent-dependent system – An arrangement of objects in which order does not matter
6. Permutation – Two or more algebraic sentences joined by the word and
7. Combination – A system with no solutions
8. Linear programming - A solution set to a system of inequalities
1. Where are the solutions to a system located?
2. Suppose one equation of a system is in slope-intercept form and the other is in standard form. Which method presented in this chapter would be the most effective to solve this system? Why?
3. Is (–3, –8) a solution to {7x4y=11x+2y=19\begin{align*}\begin{cases} 7x-4y=11\\ x+2y=-19 \end{cases}\end{align*}?
4. Is (–1, 0) a solution to {y=08x+7y=8\begin{align*}\begin{cases} y=0\\ 8x+7y=8 \end{cases}\end{align*}?

Solve the following systems by graphing.

1. {y=2y=6x+1\begin{align*}\begin{cases} y=-2\\ y=-6x+1 \end{cases}\end{align*}
2. {y=313xx+3y=4\begin{align*}\begin{cases} y=3-\frac{1}{3} x\\ x+3y=4 \end{cases}\end{align*}
3. {y=12x64y=2x24\begin{align*}\begin{cases} y=\frac{1}{2} x-6\\ 4y=2x-24 \end{cases}\end{align*}
4. {y=45x+7y=25x+1\begin{align*}\begin{cases} y=-\frac{4}{5} x+7\\ y=\frac{2}{5} x+1 \end{cases}\end{align*}
5. x=2y=4y=12x+3\begin{align*}\begin{cases} x=2\\ y=4\\ y=\frac{1}{2} x+3 \end{cases}\end{align*}

Solve the following systems by substitution.

1. {y=2x7y+7=4x\begin{align*}\begin{cases} y=2x-7\\ y+7=4x \end{cases}\end{align*}
2. {y=3x+22y=2x+16\begin{align*}\begin{cases} y=-3x+22\\ y=-2x+16 \end{cases}\end{align*}
3. {y=313xx+3y=4\begin{align*}\begin{cases} y=3-\frac{1}{3} x\\ x+3y=4 \end{cases}\end{align*}
4. {2x+y=10y=x+14\begin{align*}\begin{cases} 2x+y=-10\\ y=x+14 \end{cases}\end{align*}
5. {y+19=7xy=2x9\begin{align*}\begin{cases} y+19=-7x\\ y=-2x-9 \end{cases}\end{align*}
6. {y=05x=15\begin{align*}\begin{cases} y=0\\ 5x=15 \end{cases}\end{align*}
7. {y=313xx+3y=4\begin{align*}\begin{cases} y=3-\frac{1}{3}x\\ x+3y=4 \end{cases}\end{align*}
8. {7x+3y=3y=8\begin{align*}\begin{cases} 7x+3y=3\\ y=8 \end{cases}\end{align*}

Solve the following systems using elimination.

1. {2x+4y=142x+4y=8\begin{align*}\begin{cases} 2x+4y=-14\\ -2x+4y=8 \end{cases}\end{align*}
2. {6x9y=276x8y=24\begin{align*}\begin{cases} 6x-9y=27\\ 6x-8y=24 \end{cases}\end{align*}
3. {3x2y=02y3x=0\begin{align*}\begin{cases} 3x-2y=0\\ 2y-3x=0 \end{cases}\end{align*}
4. {4x+3y=28x+3y=14\begin{align*}\begin{cases} 4x+3y=2\\ -8x+3y=14 \end{cases}\end{align*}
5. {8x+8y=86x+y=1\begin{align*}\begin{cases} -8x+8y=8\\ 6x+y=1 \end{cases}\end{align*}
6. {7x4y=11x+2y=19\begin{align*}\begin{cases} 7x-4y=11\\ x+2y=-19 \end{cases}\end{align*}
7. {y=2x14x+6y=10\begin{align*}\begin{cases} y=-2x-1\\ 4x+6y=10 \end{cases}\end{align*}
8. {x6y=202y3x=12\begin{align*}\begin{cases} x-6y=20\\ 2y-3x=-12 \end{cases}\end{align*}
9. {4x+4y=08x8y=0\begin{align*}\begin{cases} -4x+4y=0\\ 8x-8y=0 \end{cases}\end{align*}
10. \begin{align*}\begin{cases} -9x+6y=-27\\ -3x+2y=-9 \end{cases}\end{align*}

Graph the solution set to each system of inequalities.

1. \begin{align*}\begin{cases} y>-\frac{3}{5} x-5\\ y\ge-2x+2 \end{cases}\end{align*}
2. \begin{align*}\begin{cases} y>\frac{13}{8} x+8\\ y\ge \frac{1}{4} x-3 \end{cases}\end{align*}
3. \begin{align*}\begin{cases} y\le \frac{3}{5} x-5\\ y\ge-2x+8 \end{cases}\end{align*}
4. \begin{align*}\begin{cases} y\le-\frac{7}{5} x-3\\ y\ge \frac{4}{5} x+4 \end{cases}\end{align*}
5. \begin{align*}\begin{cases} x<5\\ y\ge\frac{9}{5} x \end{cases}\end{align*} \begin{align*}-2\end{align*}

Write a system of inequalities for the regions below.

1. Yolanda is looking for a new cell phone plan. Plan A charges $39.99 monthly for talking and$0.08 per text. Plan B charges $69.99 per month for an “everything” plan. 1. At how many texts will these two plans charge the same? 2. What advice would you give Yolanda? 1. The difference of two numbers is –21.3. Their product is –72.9. What are the numbers? 2. Yummy Pie Company sells two kinds of pies: apple and blueberry. Nine apple pies and 6 blueberry pies cost$126.00. 12 apples pies and 12 blueberry pies cost $204.00. What is the cost of one apple pie and 2 blueberry pies? 3. A jet traveled 784 miles. The trip took seven hours, traveling with the wind. The trip back took 14 hours, against the wind. Find the speed of the jet and the wind speed. 4. A canoe traveling downstream takes one hour to travel 7 miles. On the return trip, traveling against current, the same trip took 10.5 hours. What is the speed of the canoe? What is the speed of the river? 5. The yearly musical production is selling two types of tickets: adult and student. On Saturday, 120 student tickets and 45 adult tickets were sold, bringing in$1,102.50 in revenue. On Sunday, 35 student tickets and 80 adult tickets were sold, bringing in $890.00 in revenue. How much was each type of ticket? 6. Rihanna makes two types of jewelry: bracelets and necklaces. Each bracelet requires 36 beads and takes 1 hour to make. Each necklace requires 80 beads and takes 3 hours to make. Rihanna only has 600 beads and 20 hours of time. 1. Write the constraints of this situation as a system of inequalities. 2. Graph the feasible region and locate its vertices. 3. Rihanna makes$8.00 in profit per bracelet and $7.00 in profit per necklace. How many of each should she make to maximize her profit? 7. A farmer plans to plant two type of crops: soybeans and wheat. He has 65 acres of available land. He wants to plant twice as much soybeans as wheat. Wheat costs$30 per acre and soybeans cost 30 per acre. 1. Write the constraints as a system of inequalities. 2. Graph the feasible region and locate its vertices. 3. How many acres of each crop should the farmer plant in order to minimize cost? 8. How many ways can you organize 10 items on a shelf? 9. Evaluate 5! 10. Simplify \begin{align*}\frac{100!}{97!}\end{align*}. 11. How many ways can a football team of 9 be arranged if the kicker must be in the middle? 12. How many one-person committees can be formed from a total team of 15? 13. How many three-person committees can be formed from a total team of 15? 14. There are six relay teams running a race. How many different combinations of first and second place are there? 15. How many ways can all six relay teams finish the race? 16. Evaluate \begin{align*}\binom{14}{12}\end{align*}. 17. Evaluate \begin{align*}\binom{8}{8}\end{align*} and explain its meaning. 18. A baked potato bar has 9 different choices. How many potatoes can be made with four toppings? 19. A bag contains six green marbles and five white marbles. Suppose you choose two without looking. What is the probability that both marbles will be green? 20. A principal wants to make a committee of four teachers and six students. If there are 22 teachers and 200 students, how many different committees can be formed? ### Systems of Equations and Inequalities; Counting Methods Test 1. True or false? A shorter way to write a permutation is \begin{align*}\binom {n}{k} \end{align*}. 2. Is (–17, 17) a solution to \begin{align*}\begin{cases} y=\frac{1}{17} x+18\\ y=-\frac{21}{17} x-4 \end{cases}\end{align*}? 3. What is the primary difference between a combination and a permutation? 4. An airplane is traveling a distance of 1,150 miles. Traveling against the wind, the trip takes 12.5 hours. Traveling with the wind, the same trip takes 11 hours. What is the speed of the plane? What is the speed of the wind? 5. A solution set to a system of inequalities has two dashed boundary lines. What can you conclude about the coordinates on the boundaries? 6. What does \begin{align*}k\end{align*} have to be to create a dependent-consistent system? \begin{align*}\begin{cases} 5x+2y=20\\ 15x+ky=60 \end{cases}\end{align*} 7. Joy Lynn makes two different types of spring flower arrangements. The Mother’s Day arrangement has 8 roses and 6 lilies. The Graduation arrangement has 4 roses and 12 lilies. Joy Lynn can use no more than 120 roses and 162 lilies. If each Mother’s Day arrangement costs32.99 and each Graduation arrangement costs \$27.99, how many of each type of arrangement should Joy Lynn make to make the most revenue?
8. Solve the system \begin{align*}\begin{cases} -6x+y=-1\\ -7x-2y=2 \end{cases}\end{align*}.
9. Solve the system\begin{align*}\begin{cases} y=0\\ 8x+7y=8 \end{cases}\end{align*}.
10. Solve \begin{align*}\begin{cases} y=x+8\\ y=3x+16 \end{cases}\end{align*}.
11. How many solutions does the following system have? \begin{align*}\begin{cases} y=-2x-2\\ y=-2x+17 \end{cases}\end{align*}
12. The letters to the word VIOLENT are placed into a bag.
1. How many different ways can all the letters be pulled from the bag?
2. What is the probability that the last letter will be a consonant?
1. Suppose an ice cream shop has 12 different topping choices for an ice cream sundae. How many ways can you choose 5 of the 12 toppings?
2. A saleswoman must visit 13 cities exactly once without repeating. In how many ways can this be done?

#### Texas Instruments Resources

In the CK-12 Texas Instruments Algebra I FlexBook, there are graphing calculator activities designed to supplement the objectives for some of the lessons in this chapter. See http://www.ck12.org/flexr/chapter/9617.

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