chapter 4 section 2. objectives 1 copyright © 2012, 2008, 2004 pearson education, inc. solving...

Chapter 4 Section 2

Objectives

1

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Solving Systems of Linear Equations by Substitution

Solve linear systems by substitution.

Solve special systems by substitution.

Solve linear systems with fractions and decimals by substitution.

4.2

2

3


Objective 1


Slide 4.2-3



Graphing to solve a system of equations has a serious drawback. It is

difficult to find an accurate solution, such as from a graph.

One algebraic method for solving a system of equations is the

substitution method.

This method is particularly useful for solving systems in which one

equation is already solved, or can be solved quickly, for one of the

variables.

1 5, ,

3 6

Slide 4.2-4


2 7 12

2

x y

x y

Solution:

The solution set found by the substitution method will be the same as the solution found by graphing. The solution set is the same; only the method is different. A system is not completely solved until values for both x and y are found.

Solve the system by the substitution method.

2 72 12yy 4 7 12y y

3 1

3 3

2y

4y

2x y

8x

42x

8, 4

Slide 4.2-5

EXAMPLE 1 Using the Substitution Method


Solving a Linear System by Substitution

Step 1: Solve one equation for either variable. If one of the variables has coefficient 1 or −1, choose it, since it usually makes the substitution method easier.

Step 2: Substitute for that variable in the other equation. The result should be an equation with just one variable.

Step 3: Solve the equation from Step 2.

Step 4: Substitute the result from Step 3 into the equation from Step 1 to find the value of the other variable.

Step 5: Check the solution in both of the original equations. Then write the solution set.

Slide 4.2-6

Solve linear systems by substitution. (cont’d)


2 7 12

3 2

x y

x y

Solution:

2 73 2 12yy 6 4 7 12y y


6 3 126 6y 3 1

3 3

8y

6y

63 2x 3 12x 15x

15, 6

Be careful when you write the ordered-pair solution of a system. Even though we found y first, the x-coordinate is always written first in the ordered pair.

Slide 4.2-7



Solution:

411 1x y

Use substitution to solve the system.

4 1x y

2 14 5 11 yy 28 1 22 5 1y y

13 1

1

3

13 3y

1y

3, 1

4 11x

1 4

2 5 11

x y

x y

4 1x 3x

Slide 4.2-8



Objective 2


Slide 4.2-9



Recall from Section 4.1 that systems of equations with graphs that

are parallel lines have no solution. Systems of equations with graphs

that are the same line have an infinite number of solutions.

Slide 4.2-10


Solution:

86 841 2 xx

Use substitution to solve the system.

16 16 8 8x x 8 8

8 4

16 2 8

y x

x y

Since the statement is false, the solution set is Ø.

It is a common error to give “false” as the solution of an inconsistent system. The correct response is Ø.

Slide 4.2-11

EXAMPLE 4 Solving an Inconsistent System by Substitution


Since the statement is true every solution of one equations is also a solution to the other, so the system has an infinite number of solutions and the solution set is {(x,y)|x + 3y = −7}.


3 7

4 12 28

x y

x y

Solution:

3 33 7y yx y 7 3x y

4 127 3 28yy 28 12 12 28y y

28 28

It is a common error to give “true” as the solution of a system of dependent equations. Remember to give the solution set in set-builder notation using the equation in the system that is in standard form with integer coefficients that have no common factor (except 1).

Slide 4.2-12

EXAMPLE 5 Solving a System with Dependent Equations by Substitution


Objective 3

Solve linear systems with fractions and decimals by substitution.

Slide 4.2-13



1 1 1

2 3 31

2 22

x y

x y

Solution:

4 44 4y yx y

21

22

22x y

4 4x y

1 1 1

2 36

36 x y

3 2 2x y 2 243 4 yy

12 12 12 122 2y y

10 1

1

0

10 0y

1y

4 14x 4 4x

0x 0, 1

Slide 4.2-14

EXAMPLE 6 Using the Substitution Method with Fractions as Coefficients



0.1 0.3 0.1x y

Solution:

21 3x 7x

7, 2Slide 4.2-15

EXAMPLE 7 Using the Substitution Method with Decimals as Coefficients

0.2 1.2 1x y

0.2 1.210 10 10 1x y

0.2 1. 1 110 2 0x y 0.1 0.3 .10 0 110x y

2 12 10x y 0.1 0.310 10 0 .011x y

2y 6 12y

2 6 12 10y y 1 32( ) 12 10y y

1 3x y 3 1x y

chapter 4 section 2. objectives 1 copyright © 2012, 2008, 2004 pearson education, inc. solving...

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