Chapter 4 SOLVING SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES

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1 Chapter 4 SOLVING SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES 4.1 Introduction to Systems of Linear Equations: Solving by Graphing Objectives A Decide whether an ordered pair is a solution of a system of linear equations in two variables. B Determine the number of solutions of a system of linear equations. C Solve a system of linear equations by graphing. D Solve applied problems involving systems of linear equations. MATHEMATICALLY SPEAKING In exercises 1 4 fill in the blank with the most appropriate term or phrase from the given list. are parallel graph set of equations coincide system of equations solution 1. If the system has infinitely many solutions, the lines. 2. A(n) of a system of two equations in two variables is an ordered pair of numbers that makes both equations in the system true. 3. If the system has no solutions, the lines. 4. A(n) is a group of two or more equations solved simultaneously. EXAMPLES AND PRACTICE Review this example for Objective A: Decide whether an ordered pair is a solution of a system of linear equations in two variables. 1. Is (2, 1) a solution of the system? x+ y = 2 3x y = 4 Practice: 1. Is (2, 0) a solution of the system? 2x+ y = 4 x y = 2 Substitute the x-coordinate 2 for x and the y-coordinate 1 for y in the equations and check if both equations are true.? x+ y = = 2 3 = 2 False 3x y = 4 3(2) 1= 4 5= 4 False? Copyright 2014 Pearson Education, Inc. 119

2 The ordered pair (2, 1) is not a solution of the system because it is does not satisfy both equations. Review this example for Objective B: Determine the number of solutions of a system of linear equations. 2. Determine the number of solutions for the system graphed. Practice: 2. Determine the number of solutions for the system graphed. The two lines in this graph intersect at a single point. So the system has one solution. Review this example for Objective C: Solve a system of linear equations by graphing. 3. Solve the following system by graphing: y = x 1 x+ y = 5 Practice: 3. Solve the following system by graphing: y = 2x x+ y = 2 Graph each linear equation by using the x- and y-intercept method and then sketch the line that passes through these points. y = x 1 x+ y = 5 x y x y Plot the points, and graph both equations. 120 Copyright 2014 Pearson Education, Inc.

3 The lines appear to intersect at the point ( 2, 3). Check: Confirm that ( 2, 3) is the solution by substituting these values into the original equations:? y = x 1 3= = 3 True x+ y = ( 3) = 5 5 = 5 True So ( 2, 3) is the solution of the system.? Review this example for Objective D: Solve applied problems involving systems of linear equations. 4. A plane flying with a tail wind, flew at a speed of 550 mph, relative to the ground. When flying against the tail wind, it flew at a speed of 500 mph. Express these relationships as equations. Find the speed of the plane in calm air and the speed of the wind. Let x represent the speed of the plane, and let y represent the speed of the wind. The given information can be expressed as: x+ y = 550 x y = 500 Choose an appropriate scale and graph both equations. Practice: 4. Bill the plumber charges $80 for a house call and then $45 per hour for labor. Sue the plumber charges $65 for a house call and then $50 per hour for labor. Write a cost equation for each plumber, where y is the total cost of plumbing repairs and x is the number of hours of labor. For how many hours of labor would Bill and Sue charge the same amount? Copyright 2014 Pearson Education, Inc. 121

4 The lines appear to intersect at (525, 25). Verify that this is the solution by substituting the values into both of the equations. The solution is the speed of the plane in calm air is 525 mph, and the speed of the wind is 25 mph. ADDITIONAL EXERCISES Objective A Decide whether an ordered pair is a solution of a system of linear equations in two variables. Indicate whether each ordered pair is or is not a solution to the given system. 1. 5x 3y = 3 4x 2y = 10 for (10, 3) 2. 2x+ 2y = 8 for (1, 3) 6x y = Copyright 2014 Pearson Education, Inc.

5 Objective B Determine the number of solutions of a system of linear equations. For each system graphed, determine the number of solutions Objective C Solve a system of linear equations by graphing. Solve by graphing. 5. x y = 6 x+ y = 0 6. x+ 2y = 2 x+ y = 3 7. x+ y = 2 y = x 2 8. x 2y = 3 y = 2x 3 Copyright 2014 Pearson Education, Inc. 123

8 Objective D Solve applied problems involving systems of linear equations. Solve. 17. A movie fan rented 5 films at a local video store. The daily rental charge was $3 on some films and $5 on others. If the total rental charge was $17, how many $5 films were rented? 126 Copyright 2014 Pearson Education, Inc.

9 Chapter 4 SOLVING SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES 4.2 Solving Systems of Linear Equations by Substitution Objectives A Solve a system of linear equations by substitution. B Solve applied problems involving systems of linear equations. EXAMPLES AND PRACTICE Review this example for Objective A: Solve a system of linear equations by substitution. 1. Solve by substitution: 3x+ 2y = 6 x+ y = 1 Practice: 1. Solve by substitution: x+ 2y = 3 3x+ 6y = 4 First solve for x or y in either of the equations. Let s solve for y in the second equation. x+ y = 1 y = x 1 Next, substitute the expression x 1 for y in the first equation and solve for x. 3x+ 2y = 6 3x+ 2( x 1) = 6 3x 2x 2= 6 x 2= 6 x = 4 Solve for y by substituting 4 for x in the original second equation. x+ y = 1 4+ y = 1 y = 3 So the solution is ( 4, 3). Check this in the original system. Copyright 2014 Pearson Education, Inc. 127

10 Review this example for Objective B: Solve applied problems involving systems of linear equations. 2. During a sale, a store sells red-dot items at a 35% discount and yellow-dot items at 25% discount. A shopper bought redand yellow-dot items with a combined regular price of $57. If the total discount was $16.71, how much did the shopper spend on each kind of item? Practice: 2. On a particular airline route, a full-price coach ticket costs $350 and a discounted coach ticket costs $250. On one of these flights, there were 158 passengers in coach, which resulted in a total ticket income of $49,600. How many full-price tickets were sold? Let r represent the regular price of the red-dot items purchased and y represent the regular price of the yellow-dot items purchased. The equation representing the combined regular price of the items purchased is r+ y = 57 The equation representing the total discount is 0.35r+ 0.25y = The system of equations is r+ y = r+ 0.25y = Solve the first equation for y: r+ y = 57 y = 57 r Now substitute 57 r for y in the second equation: 0.35r+ 0.25y = r+ 0.25(57 r) = r r = r = 2.46 r = 24.6 Solve for y by substitution 24.6 for r in the first original equation. r+ y = y = 57 y = 32.4 The solution of the system is r = 24.6 and y = In the context of this 128 Copyright 2014 Pearson Education, Inc.

11 problem, the shopper spent 0.65(24.6) = $15.99 on red-dot items and 0.75(32.4) = $24.30 on yellow-dot items. ADDITIONAL EXERCISES Objective A Solve a system of linear equations by substitution. Solve by substitution and check. 1. 6x y = 32 y = x y = 3x+ 10 y = 2x Copyright 2014 Pearson Education, Inc. 129

14 11. 7x+ 8y = 0 2x y = x y = 6 5x+ y = 6 Objective B Solve applied problems involving systems of linear equations. Solve. 13. A bottle of fruit juice contains 10% water. How much water must be added to this bottle to produce 7 L of fruit juice that is 55% water? 132 Copyright 2014 Pearson Education, Inc.

15 14. A student took out two loans totaling $8000. She borrowed the maximum amount she could at 5% and the remainder at 6% interest per year. At the end of the first year, she owed $430 in interest. How much was loaned at each rate? 15. A $30,000 investment was split so that part was invested at 8% annual rate of interest and the rest at 10%. If the total annual earnings were $2620, how much money was invested at each rate? Copyright 2014 Pearson Education, Inc. 133

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17 Chapter 4 SOLVING SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES 4.3 Solving Systems of Linear Equations by Elimination Objectives A Solve a system of linear equations by elimination. B Solve applied problems involving systems of linear equations. EXAMPLES AND PRACTICE Review this example for Objective A: Solve a system of linear equations by elimination. 1. Solve the following system by the elimination method. x+ y = 3 x+ y = 7 Practice: 1. Solve the following system by the elimination method. x+ y = 2 2x y = 1 Since the coefficients of the x-terms in the two equations are opposites, the x- terms are eliminated if we add the equations. x+ y = 3 x+ y = y = 4 2y = 4 y = 2 Substitute 2 for y in either of the original equations. Substituting in the first equations we get: x+ y = 3 x + 2= 3 x = 5 So x = 5 and y = 2. That is, the solution is ( 5, 2). Check the solution in both original equations. Copyright 2014 Pearson Education, Inc. 135

18 Review this example for Objective B: Solve applied problems involving systems of linear equations. 2. To enter a zoo, adult visitors must pay $8, whereas children and seniors pay only half price. On one day the zoo collected a total of $1580. If the zoo had 246 visitors that day, how many halfprice admissions and how many fullprice admissions did the zoo collect? Practice: 2. A crew team rows in a river with a current. When the team rows with the current, the boat travels 16 miles in 2 hours. Against the current, the team rows 8 miles in the same amount of time. At what speed does the team row in still water? Let f represent the number of full-priced admissions, and let h represent the number of half-priced admissions. We must solve the following system: 8 f + 4h= 1580 f + h= 246 Eliminate the h-terms by multiplying the second equation by 4 and adding the equations. 8 f + 4h= 1580 f + h= f 4h= 984 Add the equations: 8 f + 4h= f 4h= f = 596 f = 149 Substitute 149 for f in the original second equation and solve for h. f + h= h = 246 h = 97 So on that particular day the zoo collected 149 full-priced admissions and 97 half-priced admissions. 136 Copyright 2014 Pearson Education, Inc.

19 ADDITIONAL EXERCISES Objective A Solve a system of linear equations by elimination. Solve. 1. m+ n= 4 2m n= x+ 3y = 4 4x+ 6y = 3 3. x+ 2y = 1 2x+ 3y = x+ y = 3 x y = x 3y = 5 4x 6y = x 2y = 2 5x y = 5 Copyright 2014 Pearson Education, Inc. 137

22 Objective B Solve applied problems involving systems of linear equations. Solve. 15. The annual salaries of a congressman and a senator total $288,900. If the senator makes $41,300 more than the congressman, find each of their salaries. 16. A novelty shop sells some embroidered scarves for $10 each and others for $14 each. A customer pays $92 for 8 scarves. How many scarves at each price did she buy? 140 Copyright 2014 Pearson Education, Inc.

23 Chapter 4 SOLVING SYSTEMS OF LINEAR EQUATIONS AND INEQUALITIES 4.4 Solving Systems of Linear Inequalities Objectives A Solve a system of linear inequalities by graphing. B Solve applied problems involving systems of linear inequalities. MATHEMATICALLY SPEAKING In exercises 1 2 fill in the blank with the most appropriate term or phrase from the given list. shaded regions triples boundary line pairs simultaneous both coordinate plane a system of 1. A solution to a system of two linear inequalities is a point that lies in both of the graph. 2. The graph of the inequality 2x+ 3y 1 includes the. EXAMPLES AND PRACTICE Review this example for Objective A: Solve a system of linear inequalities by graphing. 1. Graph the solutions of the system: y 3x 1 y< x y> x 1 2 Practice: 1. Graph the solutions of the system: y 2x+ 4 1 y< x 1 2 y 3x+ 2 Begin by graphing each inequality on the same coordinate plane. Graph each boundary line, then for each inequality, shade the half-plane that contains its solutions. The solutions of the system are all the points that lie in the intersection of the shaded regions. Note that points on the line y= 3x are solutions but points 1 on the lines y= x+ 2 and 3 Copyright 2014 Pearson Education, Inc. 141

24 1 y= x 1 are not. 2 Review this example for Objective B: Solve applied problems involving systems of linear inequalities. 2. The yearbook staff is selling pages of space to students for the upcoming yearbook. A full page costs $125 and a half page costs $75. The yearbook must raise at least $2000 in revenue from these pages. More students bought full pages than half pages. a. Express this information as a system of inequalities. b. Graph the system. c. Give an example of the number of full pages and the number of half pages the yearbook staff may have sold under the given conditions. a. Let x represent the number of full pages sold, and y represent the number of half pages sold. Then the system to be solved is 125x+ 75y 2000 x> y which can be rewritten as 5 80 y x+ 3 3 y< x Practice: 2. Carlo and Anita make mailboxes and toys in a craft shop. Each mailbox, x, requires 1 hr of work from Carlo and 1 hr from Anita. Each toy, y, requires 1 hr of work from Carlo and 3 hr from Anita. Carlo can work no more than 7 hr per week and Anita can work no more than 15 hr per week. a. Express this information as a system of inequalities. b. Graph the system. c. What does the solution region represent? 142 Copyright 2014 Pearson Education, Inc.

25 b. Solve by graphing. The solutions of the system are points that lie in the intersection of the two shaded regions, including part of the 5 80 boundary line y= x c. The integer solutions in the shaded region represent all possible numbers of full pages and half pages the yearbook staff may have sold under the given conditions. One possible combination is 20 full pages and 10 half pages. ADDITIONAL EXERCISES Objective A Solve a system of linear inequalities by graphing. Solve by graphing. 1. y< 8x 3 y< 4x y> 4x 3 y< 2x+ 5 Copyright 2014 Pearson Education, Inc. 143

28 15. y 2x+ 4 y> 2x 1 x 2< 0 Objective B Solve applied problems involving systems of linear inequalities. Solve. 16. A college student works in both the school cafeteria and library. She works no more than 10 hours per week in the cafeteria and no more than 17 hours per week in the library. She must work at least 20 hours each week. a. Express this information as a system of inequalities. b. Graph the system. c. How many hours can she work in the library if she works 8 hours in the cafeteria in one week? 146 Copyright 2014 Pearson Education, Inc.