Intermediate Algebra (11th edition)

CHAPTER 4 Summary 255

Solving a Linear System by Elimination

Step 1 Write both equations in standard form.

Step 2 Make the coefficients of one pair of variable
terms opposites.

Step 3 Add the new equations. The sum should be an
equation with just one variable.

Step 4 Solve the equation from Step 3.

Step 5 Find the value of the other variable by substi-
tuting the result of Step 4 into either of the
original equations.

Step 6 Check the ordered-pair solution in bothof the
originalequations. Then write the solution set.

If the result of the addition step (Step 3) is a false
statement, such as the graphs are parallel lines
andthere is no solution. The solution set is.

If the result is a true statement, such as the graphs
are the same line, and an infinite number of ordered pairs
are solutions. The solution set is written in set-builder
notation as

,

where a form of the equation is written in the blank.

51 x,y 2 | 6

0 =0,

0

0 =4,

Solve by elimination.

(1)

(2)

To eliminate y, multiply equation (1) by 3 and add the result to

equation (2).

3 times equation (1)

(2)

Add.

Divide by 17.

Let in equation (1), and solve for y.

Check to verify that is the solution set.

Solution set:

0 = 0 Solution set: 51 x,y 2 |x- 2 y= 66

• x+ 2 y=- 6

x- 2 y= 6

0 = 4 0

• x+ 2 y=- 2

x- 2 y= 6

511 ,- 326

y=- 3

5112 +y= 2

x= 1

x= 1

17 x = 17

2 x- 3 y= 11

15 x+ 3 y= 6

2 x- 3 y= 11

5 x+ y= 2

CONCEPTS EXAMPLES

4.2 Systems of Linear Equations in Three

Variables

Solving a Linear System in Three Variables

Step 1 Select a focus variable, preferably one with
coefficient 1 or and a working equation.

Step 2 Eliminate the focus variable, using the working
equation and one of the equations of the system.

Step 3 Eliminate the focus variable again, using the
working equation and the remaining equation
of the system.

- 1,

Solve the system.

(1)

(2)

(3)

We choose zas the focus variable and (2) as the working equation.

Add equations (1) and (2).

(4)

Add equations (2) and (3).

3 x+ 2 y= 13 (5)

2 x+ 3 y= 12

2 x+ y-z= 7

x+ y+z= 6

x+ 2 y-z= 6

(continued)

Step 4 Solve the system of two equations in two
variables formed by the equations from Steps 2
and 3.

Use equations (4) and (5) to eliminate x.

Multiply (4) by

Multiply (5) by 2.

Add.

y= 2 Divide by -5.

• 5 y=- 10

6 x+ 4 y= 26

• 6 x- 9 y=- 36 - 3.