9.1 Solving Systems of Linear Equations by Substitution BobsMathClass.Com Copyright © 2010 All...

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9.1 Solving Systems of Linear Equations by Substitution

BobsMathClass.Com Copyright © 2010 All Rights Reserved.

Two linear equations in two variables will be considered together to form a system of two linear equations in two variables.

Examples: x 2y 8x y 1

3x 2y 6

2x 5y 10 6x 7y 21

5x y 18

To solve a system (such as the above examples) means to find all of the ordered pairs that simultaneously satisfy both equations in the system. For example, graphing the two equations on the same set of axes will show the intersection of the two lines. x axis

y axis

(2,3)

x y 1

x 2y 8

The ordered pair which is theintersection of the two lines is the solution of the system. Therefore {(2, 3)} is the solution set of the system.

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Since the graph of a linear equation is a straight line, there are three possibilities for the solution of a system of two linear equations.

Case 1: Case 2: Case 3:

one solution no solution infinitely many solutions

Case 1: The graphs of the two equations are two lines intersecting in one point. There is exactly one solution, and the system is called a consistent system.

Case 2: The graphs of the two equations are parallel lines. There is no solution, and the system is called an inconsistent system.

Case 3: The graphs of the two equations are the same line and there are infinitely many solutions of the system. Any pair of real numbers that satisfies one of the equations also satisfies the other equation, and we say that the equations are dependent.

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One method of solving a system of equations is by graphing. However this method becomes impractical. Fortunately we have other methods for solving a system of equations.

The Substitution Method

Procedure for solving a system of two equations with two unknowns.

Step 1. Solve one of the equations for one variable in terms of the other.

Step 2. Substitute that expression obtained in Step 1 into the other equation. The result should be an equation with just one variable.

Step 3. Solve the equation from Step 2.

Step 4. Use the solution obtained in Step 3, along with the expression obtained in Step 1 to determine the solution of the system.

Step 5. Check the solution in both of the given equations.

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Solution:

Example 1. Solve the system by substitution: 2x 3y 2x 3y 10

Choose an equation to solve for one variable. Solving for x in the second equation would be easier.Now Substitute 3y +10 for ‘x’ in the first equation. This will give an equation with only one variable. Then solve for y.

x 3y 10 x 3y 10

(3y 102 y) 3 2

2x 3y 2

6y 20 3y 2 9y 18

y 2

Next, substitute –2 for y in the equation x=3y+10 to obtain the value for the variable x.

( 2x 3 10) x 4

The solution set is 4, 2

Then perform the check in both equations. 2(4) 3( 2) 2 4 3( 2) 10 8 ( 6) 2 4 6 10

Solve the system by substitution: x 3y 254x 5y 19

Your Turn Problem #1

Answer: 4, 7

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Solution:

Example 2. Solve the system by substitution: 4x 3y 73x 2y 6

Let’s solve for x in the second equation. (Either variable can be chosen in either equation.)

3x 2y 6

x2y 6

3

Next, substitute 3 for y in the equation solved for x.

Now Substitute for ‘x’ in the first equation and solve.

2y 63 2y 6

34 3y 7

4 2y 6 9y 21

y 3(32

x3) 6

x 4The solution set is 4,3 .

Perform check in both equations to verify answer. (Not shown.)

Solve the system by substitution: 3x 5y 15x 2y 11


Answer: 3, 2

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Solution:

Example 3. Solve the system by substitution: 3x y 66x 2y 8

Solve for y in the first equation.Now Substitute 3x – 6 for ‘y’ in the second equation and solve. 3x 66x 2 8

6x 6x 12 8

12 8

3x y 6

y 3x 6

y 3x 6

Since 12 8, there is no solution. The system is inconsistent with an empty solution set. {}

Answer:

Solve the system by substitution: 4x y 9y 15 4x


Answer:

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Solution:

Example 4. Solve the system by substitution: 3x 2y 6

3y x 3

2

y is already solved for in the second equation.

Substitute for 'y' in the in the first equation and sol3

vx 32

e.

3x 33 6

2x 2

6 6

3x 3x 6 6

This true statement, 6=6, indicates that a solution of one of the equations is also a solution of the other, so the solution set is an infinite set of ordered pairs. The two equations are dependent.

Answer:

To express the solution set, let x = k.

3Then y k 3

2

The solution set is then: 3k, k 3 k is a real number

2

Solve the system by substitution: 4x y 28x 2y 4


Answer: A dependent system

k, 4k 2 k is a real number

9.1 Solving Systems of Linear Equations by Substitution BobsMathClass.Com Copyright © 2010 All...

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