Beginning Algebra, 11th Edition

SECTION 4.3 Solving Systems of Linear Equations by Elimination^265

Notice that yhas been eliminated. The result, gives the x-value of the solution

of the given system. To find the y-value of the solution, substitute 4 for xin either of

the two equations of the system. We choose equation (1).

(1)

Let.

Subtract 4.

Check the solution, in both equations of the given system.

CHECK (1) (2)

Substitute. Substitute.

✓ True ✓ True

Since both results are true, the solution set of the system is 51 4, 1 26. NOW TRY

5 = 5 3 = 3

4 + 1  5 4 - 1  3

x+ y= 5 x - y= 3

14 , 12 ,

y= 1

4 +y= 5 x= 4

x+y= 5

NOW TRY x=4,

EXERCISE 1
Use the elimination method
to solve the system.

3 x+y= 8

x-y= 4

With the elimination method, the idea is to eliminateone of the variables. To d o

this, one pair of variable terms in the two equations must have coefficients that are

opposites (additive inverses).

Solving a Linear System by Elimination

Step 1 Write both equations in standard form,

Step 2 Transform the equations as needed so that the coefficients of

one pair of variable terms are opposites.Multiply one or both

equations by appropriate numbers so that the sum of the coeffi-

cients of either the x- or y-terms is 0.

Step 3 Addthe new equations to eliminate a variable. The sum should be

an equation with just one variable.

Step 4 Solvethe equation from Step 3 for the remaining variable.

Step 5 Substitutethe result from Step 4 into either of the original equa-

tions, and solve for the other variable.

Step 6 Checkthe solution in both of the original equations. Then write the

solution set.

Ax+ By= C.

It does not matter which variable is eliminated first. Usually, we choose the one

that is more convenient to work with.

Using the Elimination Method

Solve the system.

(1)

(2)

Step 1 Write both equations in standard form,.

Subtract 2xand 11 in equation (1).

Subtract y in equation (2).

Step 2 Because the coefficients of yare 1 and , adding will eliminate y. It is not

necessary to multiply either equation by a number.

- 1

5 x-y= 26

- 2 x+y=- 11

Ax+By=C

5 x= y+ 26

y+ 11 = 2 x

EXAMPLE 2

NOW TRY ANSWER

1. 51 3, - 126