# Chapter 8. Solving Systems of Linear Equations

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1 Chapter 8 Solving Systems of Linear Equations

2 Sec Solving Systems of Linear Equations by Graphing Definition: A system of linear equations consists of two or more linear equations with the same variables. Example: 5x + 6 y 14 2x + 5y 3

3 Sec Solving Systems of Linear Equations by Graphing Definition: A solution of a system of linear equations is an ordered pair (x,y) that makes both equations true at the same time.

4 Sec Solving Systems of Linear Equations by Graphing Example (4, -1): 5 x + 6y 14 2 x + 5y 3 5 (4) + 6( 1) 14 2 (4) + 5( 1)

5 Sec Solving Systems of Linear Equations by Graphing If (4,-1) satisfies both equations, then the graphs of both equations must pass through that point. Therefore, one method of solving linear equations is to graph each equation. If the two lines intersect, the point of intersection will be a solution to the system of equations.

6 Sec Solving Systems of Linear Equations by Graphing

7 Sec Solving Systems of Linear Equations by Graphing Observations: 1. The problem with the graphing method, of course, is that it is not more often than not difficult to determine the exact coordinates of the point of intersection. Therefore, other methods are traditionally preferred. 2. However, this method is best for understanding the nature of the problem.

8 Sec Solving Systems of Linear Equations by Graphing Example: 3x- y 4 6x-2y12 3x-y4 6x-2y12 x y x y

9 Sec Solving Systems of Linear Equations by Graphing The lines are parallel, and have no points in common.

10 Sec Solving Systems of Linear Equations by Graphing Example: 2x+ 5y1 6x+15y3 2x+5y1 6x+15y3 x y x Y The graphs of these two equations are the same line.

11 Sec Solving Systems of Linear Equations by Graphing So we have determined three types of solutions: 1. The graphs intersect at a point. A system with a solution is called consistent. 2. The graphs are parallel lines. The system of equations is called inconsistent. 3. The graphs are the same. The system of equations is called dependent.

12 Sec Solving Systems of Linear Equations by Graphing We can identify a system of equations that is inconsistent or dependent by putting the equations in slope-intercept form.

13 Sec Solving Systems of Linear Equations by Graphing Inconsistent equations: 3 x y y 3x x 2y 12 3x 4 y 6 x 12 2y y 3x 6 The inconsistent equations have the same slope.

14 Sec Solving Systems of Linear Equations by Graphing Dependent equations: x + 3 y 2 2x 6y y x + 2 6y 2x 1 y x + 2 y 1 x The dependent equations have the same slope and intercept.

15 Sec Solving Systems of Linear Equations by Graphing Class Problem: Determine if the following system of equations is consistent, inconsistent, or dependent. 2 x + 3y 4 x 2y

16 Sec Solving Systems of Linear Equations by Graphing Class Problem: Determine if the following system of equations is consistent, inconsistent, or dependent. 2 x y 4 x 2y + 4 8

17 Sec Solving Systems of Linear Equations by Substitution As we noted in the previous section, it can be difficult to determine the exact solution of system of equations by the graphing method. Therefore, a purely algebraic method can have an advantage.

18 Sec Solving Systems of Linear Equations by Substitution Example: 2x+7y-12 x-2y 2x + 7y 2( 2y) + 7 y 3y y x 8

19 Sec Solving Systems of Linear Equations by Substitution 1. Solve one of the equations for a variable in terms of the other. 2. Substitute for the solved variable in the other equation to get an equation in one variable. 3. Solve the equation in one variable 4. Substitute the value from step 3 into the equation from step 1 to get the value for the other variable. 5. Check the solution in both equations.

20 Sec Solving Systems of Linear Equations by Substitution Example: x+ 1-4y 2x-5y11 1. Solve one of the equations for a variable in terms of the other. x 4y 1

21 Sec Solving Systems of Linear Equations by Substitution Example: x+ 1-4y 2x-5y11 1. Substitute for the solved variable in the other equation to get an equation in one variable. x 4y 1 2 ( 4y 1) 5y 11

22 Sec Solving Systems of Linear Equations by Substitution Example: x+ 1-4y 2x-5y11 1. Solve the equation in one variable 2 ( 4y 1) 5y 13 y y 1

23 Sec Solving Systems of Linear Equations by Substitution Example: x+ 1-4y 2x-5y11 1. Substitute the value from step 3 into the equation from step 1 to get the value for the other variable. y x + 1 x 1 4 3

24 Sec Solving Systems of Linear Equations by Substitution Example: x+ 1-4y 2x-5y11 1. Check the solution in both equations. x + 1 4y 2 x 5y ( 1) 2 ( 3) 5( 1)

25 Sec Solving Systems of Linear Equations by Substitution Example: x+ 1-4y 2x-5y11

26 Sec Solving Systems of Linear Equations by Substitution The graphing method did show us two possible problem situations: inconsistent and dependent systems of equations. We will now look at what happens in this situation when we use the substitution method.

27 Sec Solving Systems of Linear Equations by Substitution Example: y 8x x - 2y 8 This system is inconsistent, for when we write each equation in the slope-intercept form we get: y 8x + 4 y 8x - 4

28 Sec Solving Systems of Linear Equations by Substitution Example: y 8x x - 2y 8 Solving using the substitution method: 16x 2(8x + 4) This can not be true, so we know the system is inconsistent.

29 Sec Solving Systems of Linear Equations by Substitution Example: x + 3y -7 12y -4x -28 This system is dependent, for when we write each equation in the slope-intercept form we get, for each equation: 1 7 y x 3 3

30 Sec Solving Systems of Linear Equations by Substitution Example: x + 3y -7 12y -4x y x x + 3( x ) x x Which is always true.

31 Sec Solving Systems of Linear Equations by Substitution Example using fractional coefficients: 2 x x y y 6 0

32 Sec Solving Systems of Linear Equations by Substitution Clear fractions from the first equation x + y x + 3y 36

33 Sec Solving Systems of Linear Equations by Substitution Clear fractions from the second equation x y x 3y 0 0

34 Sec Solving Systems of Linear Equations by Substitution Solve: y x y x 6 x y x y y 36 9 y 4 y

35 Sec Solving Systems of Linear Equations by Substitution Solve: 4x + 3y 36 2x 3y

36 Sec Solving Systems of Linear Equations by Elimination You will recall from Section 2.1 the Addition Property of Equality: If A, B, and C are mathematical expressions that represent real numbers, then the equations A B and A + C B + C have exactly the same solution.

37 Sec Solving Systems of Linear Equations by Elimination This can be generalized to: If A, B, C, and D are mathematical expressions that represent real numbers, then the equations A B and CD, then A + C B + D have exactly the same solution.

38 Sec Solving Systems of Linear Equations by Elimination You will recall from Section 2.1 the Multiplication Property of Equality: If A, B, and C are mathematical expressions that represent real numbers, then the equations A B and AC BC have exactly the same solution.

39 Sec Solving Systems of Linear Equations by Elimination Example: Use the elimination method to solve 3x - y 7 2x + y 3 ( 3x y) + (2x + y) x 10 x 2 y 1 3

40 Sec Solving Systems of Linear Equations by Elimination 1. Write both equations in standard form Ax + By C. 2. Multiply one or both equations by appropriate numbers so that the coefficients of x or y are opposites. 3. Add the two equations to get an equation in one variable. 4. Solve 5. Substitute the solution from step 4 into one of the original equations to get the value of the other variable. 6. Check.

41 Sec Solving Systems of Linear Equations by Elimination Example: x + 2-3y 3x - 4x + 7 2y + 7 Step 1: Rewrite equation 1 x + 2 3y x + 3y 2

42 Sec Solving Systems of Linear Equations by Elimination Example: x + 3y -2 3x - 4x + 7 2y + 7 Step 1: Rewrite equation 2 3 x 4x + 7 2y + 7 x + 7 2y + 7 x 2 y 0

43 Sec Solving Systems of Linear Equations by Elimination Example: x + 3y-2 -x - 2y 0 Step 2: Multiple one or both equations so that x or y are opposites of each other. Not necessary. Coefficients of x are equal and opposite.

44 Sec Solving Systems of Linear Equations by Elimination Example: x + 3y-2 -x - 2y 0 Step 3: Add two equations y 2

45 Sec Solving Systems of Linear Equations by Elimination Example: x + 3y-2 Step 4: Solve -x - 2y 0 Not necessary.

46 Sec Solving Systems of Linear Equations by Elimination Example: x + 3y-2 -x - 2y 0 Step 5: Substitute x x 2 y 2 ( 2) x 0 0 4

47 Sec Solving Systems of Linear Equations by Elimination Example: x + 3y-2 -x - 2y 0 Step 6: Check x + 3y 2 x 2 y ( 2) 2 4 2( 2)

48 Sec Solving Systems of Linear Equations by Elimination Example: 4x - 5y-18 3x + 2y -2 Step 1: Rewrite equations Not necessary.

49 Sec Solving Systems of Linear Equations by Elimination Example: 4x - 5y-18 3x + 2y -2 Step 2: Multiple one or both equations so that x or y are opposites of each other. 4x 5y 18 3x + 2y 2 3(4x 5y) 3( 18) 4(3x + 2y) 4( 2) 12 x + 15y 54 12x + 8y 8

50 Sec Solving Systems of Linear Equations by Elimination Example: 4x - 5y-18 3x + 2y -2 Step 3: Add two equations 12 x + 15y 12x + 8y y 46

51 Sec Solving Systems of Linear Equations by Elimination Example: 4x - 5y-18 3x + 2y -2 Step 4: Solve 23 y 46 y 2

52 Sec Solving Systems of Linear Equations by Elimination Example: 4x - 5y-18 3x + 2y -2 Step 5: Substitute 4x 5y 18 4x 5(2) 18 4x x 8 x 2

53 Sec Solving Systems of Linear Equations by Elimination Example: 4x - 5y-18 3x + 2y -2 Step 6: Check 4x 5y 18 4( 2) 5(2) x + 2y 3( 2) + 2(2)

54 Sec Solving Systems of Linear Equations by Elimination Example: 3y 8 + 4x 6x 9-2y Step 1: Rewrite equations 3 y 8 + 4x 6x 9 2y 4 x + 3y 8 6 x + 2y 9

55 Sec Solving Systems of Linear Equations by Elimination Example: -4x + 3y 8 6x + 2y 9 Step 2: Multiple one or both equations so that x or y are opposites of each other. 4 x + 3y 8 2( 4x + 3y) 8x 6y 16 2(8) 6 x + 2y 3 (6x + 2y) 18 x + 6y 9 3(9) 27

56 Sec Solving Systems of Linear Equations by Elimination Example: -4x + 3y 8 6x + 2y 9 Step 3: Add two equations 8x 6y x + 6y x 11

57 Sec Solving Systems of Linear Equations by Elimination Example: -4x + 3y 8 6x + 2y 9 Step 4: Solve 26 x x

58 Sec Solving Systems of Linear Equations by Elimination Example: -4x + 3y 8 6x + 2y 9 Step 5: Substitute Instead of working with fractions, we can return to Step 2 and eliminate x, and solve for y.

59 Sec Solving Systems of Linear Equations by Elimination Example: -4x + 3y 8 6x + 2y 9 Step 2: Multiple one or both equations so that x or y are opposites of each other. 4 x + 3y 8 6 x + 2y 9 3 ( 4x + 3y) 3(8) 2 (6x + 2y) 2(9) 12 x + 9y x + 4y 18

60 Sec Solving Systems of Linear Equations by Elimination Example: -4x + 3y 8 6x + 2y 9 Step 3: Add two equations 12 x + 9y 12 x + 4y 13 y

61 Sec Solving Systems of Linear Equations by Elimination Example: -4x + 3y 8 6x + 2y 9 Step 4: Solve 13 y y

62 Sec Solving Systems of Linear Equations by Elimination Example: -4x + 3y 8 Step 6: Check 6x + 2y 9 4 x + 3y (11) + 6(42) x + 2y (42) 9 26

63 Sec Elimination Method and Inconsistent Systems Example: 3x + y -7 6x + 2y 5 Step 1: Rewrite equations Not necessary

64 Sec Solving Systems of Linear Equations by Elimination Example: 3x + y -7 6x + 2y 5 Step 2: Multiple one or both equations so that x or y are opposites of each other. 3x + y 7 6 x + 2y 5 2(3x + y) 2( 7) 6 x 2y 14

65 Sec Solving Systems of Linear Equations by Elimination Example: 4x - 5y-18 3x + 2y -2 Step 3: Add two equations 6 x 2y 6 x + 2y Inconsistent!

66 Sec Applications Many real world problems can be solved using systems of equations. Moreover, some of the one variable problems we solved earlier are more easily worked as two variable problems.

67 Sec Solving Money Problems Example Data collected in a recent year for Major League Baseball and the National Football League indicate that for average ticket prices, 3 baseball tickets and 2 football tickets would have cost \$105.05, while 2 baseball tickets and one football ticket would have cost \$ What were the average prices for the tickets for the 2 sports?

68 Sec 8.5 Solving Money Problems Let b average price of baseball tickets f average price of football tickets 3 b + 2 f 2 b + f

69 Sec Solving Money Problems Solve using substitution 3 b + 2 f 2 b + f ( ) b + 2 b b b f f b ( 11.19)

70 Sec Applications Solving an Applied Problem by Writing a System of Equations 2. Determine what is given and what needs to be found 3. Assign variables 4. Write a system of equations 5. Solve and test for reasonableness 6. Check.

71 Sec 8.5 Solving Geometry Problems Find the measures of angles x and y x-20 x y

72 Sec Solving Geometry Problems 1. Determine what is given and what needs to be found We are given two supplementary angels; that is, the sum of their measures equals 180. We are given two vertical angles (alternate angles of intersecting lines). Their measures are equal.

73 Sec Solving Geometry Problems 1. Assign variables Already done for us; x and y.

74 Sec Solving Geometry Problems 1. Write a system of equations x + y x y

75 Sec Solving Geometry Problems 1. Solve and test for reasonableness x + y x y x + x 20 2 x x y

76 Sec Solving Geometry Problems 1. Check. x + y 180 x y

77 Sec 8.5 Solving Mixture Problems Example If a grocer has some \$4 per lb coffee and some \$8 per lb coffee that she will mix to make 50lb of \$5.60 per lb coffee. How many lbs of each should be used?

78 Sec Solving Mixture Problems 1. Determine what is given and what needs to be found We have \$4 per lb coffee and \$8 per lb coffee, and we want to make \$5.60 per lb coffee. We are need to determine the amount of \$4 coffee and \$8 coffee to mix.

79 Sec Solving Mixture Problems 1. Assign variables Let f # lbs of \$4 coffee e # lbs of \$8 coffee

80 Sec Solving Mixture Problems 1. Write a system of equations f + e 4 f + 8e ( 50)

81 Sec Solving Mixture Problems 1. Solve and test for reasonableness f + e 4 f + 8e f 4e 4 f + 8e

82 Sec Solving Mixture Problems 1. Solve and test for reasonableness 4 f 4e 4 f + 8e 4 e e f

83 Sec Solving Mixture Problems 1. Check. f + e 50 4 f + 8e ( 30) + 8( 20)

84 Sec 8.5 Solving Motion Problems Example A train travels 600 mi in the same time that a truck travels 520 mi. Find the speed of each vehicle if the trains average speed is 8 mph faster than the truck.

85 Sec Solving Motion Problems 1. Determine what is given and what needs to be found 1. Each vehicle travels the same amount of time. 2. The train travels 600 mi and the truck travels 520 mi. 3. The train travels 8 mph faster than the truck.

86 Sec Solving Motion Problems 1. Assign variables Let l speed of train t speed of train

87 Sec Solving Motion Problems 1. Write a system of equations 600 l l 520 t t t 520l

88 Sec Solving Motion Problems 1. Solve and test for reasonableness 600 t 520l l t + 8 ( 8) 600 t 520 t t 520t + 80 t 4160 t l

89 Sec Solving Motion Problems 1. Check. 600 t 520l l t ( ) ( ) 31,200 31,200 8

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