Intermediate Algebra (11th edition)

NOTEIf an equation in a system contains decimal coefficients, it is best to first clear

the decimals by multiplying by 10, 100, or 1000, depending on the number of decimal

places. Then solve the system. For example, we multiply each sideof the equation

by or 100, to get the equivalent equation

OBJECTIVE 4 Solve linear systems (with two equations and two variables)

by elimination.Another algebraic method, the elimination method,involves com-

bining the two equations in a system so that one variable is eliminated. This is done

using the following logic.

If and then

Solving a System by Elimination

Solve the system.

(1)

(2)

Notice that adding the equations together will eliminate the variable y.

(1)

(2)

Add.

Solve for x.

To find y, substitute 0 for xin either equation (1) or equation (2).

(1)

Let

Multiply.

Add.

Divide by 3.

The solution is Check by substituting 0 for xand for yin both equations

of the original system. The solution set is 51 0, - 226.

1 0, - 22. - 2

y= - 2

3 y=- 6

0 + 3 y=- 6

2102 + 3 y=- 6 x=0.

2 x+ 3 y=- 6

x= 0

6 x = 0

4 x-^3 y=^6

2 x+ 3 y=- 6

4 x- 3 y= 6

2 x+ 3 y=- 6

EXAMPLE 6

ab cd, acbd.

50 x+ 75 y=325.

102 ,

0.5x+0.75y= 3.25

SECTION 4.1 Systems of Linear Equations in Two Variables 215

NOW TRY
EXERCISE 5
Solve the system.

• 2 x+ 3 y = 1

1

10

x-

3

5

y=

2

5

Since and

A check verifies that the solution set is EA^115 ,^35 BF. NOW TRY

y= 3 a

11

5

b - 6 =

33

5

-

30

5

=

3

5

.

x=

11

y= 3 x- 6 5 ,^6 =

30

5

NOW TRY
EXERCISE 6
Solve the system.

5 x+ 2 y=- 18

8 x- 2 y= 5

NOW TRY ANSWERS
5.

6. EA-1, -^132 BF

51 - 2, - 126

NOW TRY

By adding the equations in Example 6,we eliminated the variable ybecause the

coefficients of the y-terms were opposites. In many cases the coefficients will notbe

opposites, and we must transform one or both equations so that the coefficients of

one pair of variable terms are opposites.