# Introduction to Systems of Equations

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1 Systems of Equations 1 Introduction to Systems of Equations Remember, we are finding a point of intersection x 2y 5 2x y 4 1. A golfer scored only 4 s and 5 s in a round of 18 holes. His score was 80. How many of each score did he have? 2. Tuition plus Room/Board at a local college is \$24,000. Room/Board is \$400 more than one-third the tuition. Find the tuition.

2 Systems of Equations 2 3. Mr. Trem bought 7 shirts for the coaches of his baseball team. The blue long sleeved shirts cost \$30 each and the white short sleeved shorts cost \$20 each. If he paid a total of \$160, how many of each shirt did he buy?? 4. Rob invests money, some at 10% and some at 20% earning \$20 in interest per year. Had the amounts invested been reversed, he would have received \$25 in interest. How much has he invested all together? 5. A merchant said that it did not matter whether one pair of shoes was sold for \$31 or two pairs for \$49 because the profit was the same for each sale. How much does one pair of shoes cost the merchant and what is the profit??

3 Systems of Equations 3 Determinants A determinant is an array of numbers. Each number is called an element of the array. There are 2x2 arrays which are called 2 nd order determinants 3x3 3rd Ex. a c b d a b c d e f g h i How to find the value of a determinant a c b d D = ad bc Ex Ex bc ad Cramer s Rule: ax by c aex bey ce dx ey f bdx bey bf aex bdx ce bf x(ae bd) ce bf x ce ae bf bd x c f a d b e b e The same process will find y to be y a d a d c f b e

4 Systems of Equations 4 Cramer s Rule Given a system ax by c dx ey f Replace coefficients in x column with constants to solve for x Replace coefficients in y column with constants to solve for y x c f a d b e b e y a d a d c f b e Denominators are coefficient determinants EXAMPLES 6x 7y x y 5 5x 4b 1 2x b 10

5 Systems of Equations 5 Assignment #1 2 by 2 (1) Solve each of the following systems of equations please. (a) x 2y = 3 3x y = 14 (c) 4x + 7y = 9 3x 2y = 11 (b) x 2 y = x + 2 y + 3 = (d) 1x + 1y = x + 2y = (2) There are 22 coins in a bank consisting of only nickels and quarters. The total value of the coins is \$3.90. How many nickels and how many quarters are there? (3) The treasurer of the student body reported that the receipts for the last concert totaled \$1150 and that 300 people attended. If students paid \$3 per ticket and nonstudents paid \$5 per ticket, how many students attended the concert? (4) The cost of 10 pounds of potatoes and 4 pounds of apples is \$ The cost of 4 pounds of potatoes and 8 pounds of apples cost \$ What is the cost per pound of potatoes and apples? (5) Mary and Joe went to the store. They had a total of \$22.80 to spend and came home with \$6.20. If Mary spent 2 3 of her money and Joe spent 4 5 of his money, how much did each have to begin with? (6) To get to work Al first averages 36 mph driving his car to the train station and then rides the train which averages 60mph. The entire trip takes 1 hour and 22 minutes. It costs him 15 per mile to drive his car and 6 per mile to ride the train. If the total cost of his trip is \$5.22, find the distances traveled by car and train.

6 Systems of Equations 6 Answers (1) (a) 5, 1 (c) 95, (b) 17 2, 4 (d) (2) 8 nickels; 14 quarters (3) 175 students; 125 nonstudents (4) apples \$1.25 per pound; potatoes \$2.50 per pound (5) Joe had \$10.50; Mary had \$12.30 (6) 6 miles by car; 72 miles by train

7 Systems of Equations 7 Solving 3 x 3 Linear Systems 1) 3x y z 3 7y 6z 15 5z 5 2) Labeling your equations sometimes helps keep order: 2x 14y 4z 2 4x 3y z 8 3x 5y 6z 7

8 Systems of Equations x y z 8 4x y 2z 3 3x y 2z 5 4) x 2y 3z 11 2x 3y 6 3x 3y 3z 3 Matrix Notation Matrix a rectangular array consisting of the coefficients and constants of a system of equations. 2x y 3z 0 x y 2z 1 x 2y z

9 Systems of Equations 9 1. Geometry revisited. Find the angles of a triangle with the following properties. The largest angle is equal to the sum of the other two angles. Four times the smallest angle is 20 degrees less than the largest angle. 2. A farm wants to make sure their animals consume 50 pounds of corn, 50 pounds of oats and 50 pounds of barley. They have three different feeds to purchase. How many bags of each feed should they purchase to make sure they have the correct quantities, given the breakdown of feeds: Corn Oats Barley 1 bag of Feed A 2 lbs 5 lbs 1 lb 1 bag of Feed B 1 lb 1 lb 3 lbs 1 bag of Feed C 2 lbs 0 lbs 1 lb 3. Find a quadratic function that passes through the points (2, 1), (3, 2), and (4, -1). Remember 2 y ax bx c

10 Systems of Equations 10 Assignment #2-3 by 3 systems For Problems 1-5, solve each of the following systems please. (1) x 2y = 0 2x + 5z = 7 x + y + z = 3 (2) x + y + z = 6 2x y + z = 3 3x + y + z = 8 (3) x + 2y z = 6 x y + 2z = 0 2x + 3y + z = 6 (4) 3x + y + 2z = 1 2x y + 4z = 3 5x + 2y + 6z = 4 (5) 6x + 5y 5z = 0 3x 6y + 4z = 47 2x + y 3z = 20 (6) Find a quadratic function that passes through the points (3, 5), (1, -3), and (4, 3). 2 Remember y ax bx c

11 Systems of Equations 11 Answers (1) 4, 2, 3 (2) 1, 2, 3 (3) 4, 0, 2 (4) 1, 3, 1 2 (5) 5, 4, 2 (6) 2 y 2x 12x 13

12 Systems of Equations 12 Partial Fraction Decomposition If given, we know how to combine it into However, we also need to be able to do the reverse, which is called decomposition. Find the partial fraction decomposition of Find the partial fraction decomposition of Heaviside Method:

13 Systems of Equations 13 Repeating Factors: Find the partial fraction decomposition of Non-linear factors: Find the partial fraction decomposition of Find the partial fraction decomposition of

14 Systems of Equations 14 Assignment #3 Partial Fractions For problems 1-5, find the partial-fraction decomposition for each expression (1) (2) (3) (4) (5) (6) Find the partial-fraction decomposition of and apply it to find the sum of:

15 Systems of Equations 15 Answers (1) (2) (3) (4) (5) (6)

16 Systems of Equations 16 Linear Inequalities 1. A company produces three types of skis: regular model, trick ski and slalom ski. They need to fill a customer order of 110 pairs of skis. There are two major production divisions within the company: labor and finishing. Each regular model of skis requires 2 hours of labor and 1 hour of finishing. Each trick ski model requires 3 hours of labor and 2 hours of finishing. Finally, each slalom ski model requires 3 hours of labor and 5 hours of finishing. Suppose the company has only 297 labor hours and 202 finishing hours. How many of each type ski can be made under these restrictions? 2. Graph the following linear inequalities: a. b.

17 Systems of Equations Graph the following system of inequalities: a. b. 4. A couple has invited 300 guests to their wedding. The fixed costs (such as entertainment, room rental, etc) are \$7,000. The variable costs (such as food and drink) are between \$60 and \$80 per person depending on the menu. Graph the cost of the wedding as a system of inequalities.

18 Systems of Equations Another ski producer produces two models, a regular ski and a slalom ski. Regular skis make a profit of \$35 and slalom skis make a profit of \$50. The producer has enough materials to build 500 sets of skis. He only has 1200 hours to make the skis. Slalom skis take 3 hours to make while regular skis take 2 hours. How many skis should he make to maximize profits?

19 Systems of Equations 19 Assignment #4 Linear Inequalities 1. Graph the following system of inequalities: a. c. b. d A computer science student raises \$10,000 to start a new computer business. He is going to make computers over the summer and sell them during the first days of class. He only has 90 hours to make the computers. The details of the computers are shown below: Laptop Desktop Cost of Parts \$500 \$400 Time to Assemble (hours) 5 3 Profit \$300 \$200 He estimates that demand for laptops will be at least as much as the demand for desktops (he will not sell less laptops than desktops). Set up the feasible region for the possible laptop/desktop combinations and show the optimal mix he should make in order to maximize profit.

20 Systems of Equations 20 Review Worksheet - Systems (1) Solve each of the following systems using any method please. (a) x + 3y = 6 2x 5y = 7 (b) 5x + 3y = 2 6x 7y = 10 2x + y 2z = 31 x + y + z = 1 (c) x 2y 3z = 23 (d) 3x 2y 4z = 16 x 2y + z = 3 2x y + z = 19 x + y + z = 4 x + 2y + 2z = 13 (e) x y z = 2 (f) 2x + y z = 3 x y + z = 0 x 4y + 3z = 11 (2) Al has 28 coins consisting of only dimes and quarters. He has \$5.20 in all. How many coins of each type does he have? (3) A purchase of 4 dozen apples and 6 dozen pears costs \$32. If the of apples price per dozen increases by 25% and the price of pears per dozen increases by 50%, the same order would cost \$43. Find the initial cost of a dozen apples and a dozen pears. (4) An adult ticket to a play costs \$8, while a child's ticket costs \$4. If 150 people attended the play, and the gross receipts were \$800, how many tickets of each type were sold? (5) Ed has a total of \$4000 in his savings account and in a certificate of deposit. His savings account earns 6.5% interest annually. The certificate of deposit pays 8% per year. How much does he have in each investment if his interest earnings for the year will be \$297.50? (6) To get to work, Jo first averages 40 miles per hour driving her car to the train station and then rides the train which averages 60 miles per hour. The entire trip takes 35 minutes. It costs her 20 per mile to drive to the station, and 10 per mile to ride the train. If her total cost is \$4.00, find the distances traveled by car and by train.

21 Systems of Equations 21 (7) An amateur painter is selling his works. It costs him \$5 in materials to make a pastel and he makes \$40 in profit with each sale. It costs him \$15 in materials to make a watercolor and he makes \$120 with each sale. If he only has \$180 to spend on materials and can make at most 16 paintings, how many of each should he make to maximize his profits? What is that profit? (8) Please find the partial-fraction decompositions of the following: a. c. b. (9) Graph the following system of inequalities: a. b.

22 Systems of Equations 22 Review Worksheet Answers (1) (a) 9, 19 (b) , (c) ( 10, 1, 5 ) (d) ( 4, 8, 3 ) (e) ( 1, 2, 1 ) (f) ( 3, 1, 4 ) (2) 12 dimes, 16 quarters (3) \$5 apples, \$2 pears (4) 50 adults, 100 children (5) \$1500 savings, \$2500 cd (6) 20 miles by train, 10 by car (7) 6 pastels and 10 watercolors for \$1,240 profit (8) a. b. c.

23 Systems of Equations 23 Algebra 3/Trig Extra Practice Problems A. Solve: Answers: 1. 2x - 4y x 3y 17 - x 5y 11 4x 2y 13 B. Solve using any method 3. 6x 4y 12 5x 3y (9, 4) 2. No solution , x y 3z x 2y z x 4y z ( 1,1,1) 6x - 2y z -7 3y 2z 16 10x+3y 2z 0 5. (1,4, 2) x y 4z 2 x y z 5 5x 7y ( 2,0,10) 7. 2x 7y z 0 x z 4 7. (4,0,8) 5x 3y 3z x 3y 2 4x 3y 6z 7 9y 2z x 3y 9z 2x 4y 3z 16 2x y 2z x y 5z 6-3x 2y z 1 -x 3y 8z (,, 1) (1,5,2) 12. (2,3,1) 13. 2x y 2z y 3z x y z (1,5,2) 3x 3y 9z 0 4x 6y z 1 -x 2y 2z (3, 2,1) 2x+4y 3z 16 -x 4y 5 x 2y z (2,4, 1) 16. x y 2z 9 5x 2y z 6 - x 3z x y 8 x y z 12 5x y z x y 2z 0 x 3y z 2 3x 2y z (1,2,3) 17. (2,4,6) 18. (1,0, 1)

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