Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 7.2 Solving Systems of Linear Equations...

transcript

Section 7.2

Solving Systems of

Linear Equations by

the Substitution and Addition

Method

What You Will Learn

Solving Systems of Linear Equations by Substitution Method

Solving Systems of Linear Equations by Addition Method

Procedure for Solving a System of Equations Using the Substitution Method1.Solve one of the equations for

one of the variables. If possible, solve for a variable with a numerical coefficient of 1. (This may avoid working with fractions.)

2.Substitute the expression found in step 1 into the other equation. This yields one equation in one variable.7.2-3

Procedure for Solving a System of Equations Using the Substitution Method3.Solve the equation found in

Step 2 for the variable.4.Substitute the value found in

Step 3 into the equation, you rewrote in Step 1, and solve for the remaining variable.

Example 3: An Infinite Number of Solutions, by the Substitution Method

Solve the following system of equations by substitution.

y = –2x + 32y = 6 – 4x

Example 3: An Infinite Number of Solutions, by the Substitution MethodSolutionSubstitute –2x + 3 for y in 2y = 6 – 4x

2y 6 4x

2( 2x 3) 6 4x

4x 6 6 4x

4x 4x 6 6 4x 4x

6 6 True7.2-6

Example 3: An Infinite Number of Solutions, by the Substitution MethodSolutionSince 6 = 6, the system has an infinite number of solutions.Thus, the system of equations is dependent.

Addition Method

If neither of the equations in a system of linear equations has a variable with the coefficient of 1, it is generally easier to solve the system by using the addition (or elimination) method.

Addition Method

To use this method, it is necessary to obtain two equations whose sum will be a single equation containing only one variable.

Solving a System of Equations by the Addition Method

1.If necessary, rewrite the equations so that the terms containing variables appear on one side of the equal sign and the constants appear on the other side of the equal sign.

7.2-10

Solving a System of Equations by the Addition Method

2.If necessary, multiply one or both equations by a constant(s) so that when you add the equations, the result will be an equation containing only one variable.

7.2-11

Solving a System of Equations by the Addition Method3.Add the equations to obtain a single

equation in one variable.4.Solve the equation in Step 3 for the

variable.5.Substitute the value found in Step 4

into either of the original equations and solve for the other variable.

7.2-12

Example 7: Multiplying Both Equations

Solve the following system of equations by the addition method.

3x 4y 8

2x 3y 9

7.2-13

Solution

2 3x 4y 8 3 2x 3y 9

6x 8y 16

6x 9y 27

6x 8y 16

6x 9y 27

17y 11 y

7.2-14

Example 7: Multiplying Both EquationsSolutionWe could now find x by substituting 11/17 for y in either of the original equations.Although it can be done, it gets messy.Instead, let’s solve for x by eliminating the variable y from the two original equations.

7.2-15

Solution

3 3x 4y 8 4 2x 3y 9

9x 12y 24

8x 12y 36

9x 12y 24

8x 12y 36

17x 60 x

7.2-16

Example 7: Multiplying Both EquationsSolution

The solution to the system is

7.2-17

Example 9: Modeling–A Mixture ProblemPat Williams, a pharmacist, needs 500 milliliters (ml) of a 10% phenobarbital solution. She has only a 5% phenobarbital solution and a 25% phenobarbital solution available. How many milliliters of each solution should she mix to obtain the desired solution?

7.2-18

Example 9: Modeling–A Mixture ProblemSolutionLet

x = number of ml of 5% solutiony = number of ml of 25% solution

We need 500 ml, thus

x + y = 500

7.2-19

Example 9: Modeling–A Mixture ProblemSolutionThe total amount of phenobarbital in a solution is determined by multiplying the percent of phenobarbital by the number of milliliters of solution.

0.05x 0.25y 0.10(500)

0.05x 0.25y 50

7.2-20

Example 9: Modeling–A Mixture ProblemSolutionThe system is

x y 500

0.05x 0.25y 50

5 x y 500 100 0.05x 0.25y 50

5x 5y 2500

5x 25y 5007.2-21

Example 9: Modeling–A Mixture ProblemSolution

5x 5y 2500

5x 25y 500

20y 2500

x y 500

x 125 500

Now determine x.

7.2-22

Example 9: Modeling–A Mixture ProblemSolution

Therefore, 375 ml of a 5% phenobarbital solution must be mixed with 125 ml of a 25% phenobarbital solution to obtain 500 ml of a 10% phenobarbital solution.

7.2-23

Copyright 2013, 2010, 2007, Pearson, Education, Inc. Section 7.2 Solving Systems of Linear Equations...

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