Intermediate Algebra (11th edition)

Our goal is to use the various row operations to change this matrix into one that

leads to a system that is easier to solve. It is best to work by columns.

We start with the first column and make sure that there is a 1 in the first row, first

column, position. There already is a 1 in this position.

Next, we get 0 in every position below the first. To get a 0 in row two, column

one, we add to the numbers in row two the result of multiplying each number in row

one by (We abbreviate this as ) Row one remains unchanged.

Original number times number

from row two from row one

• 2R 1 +R 2

c

1

0

- 3

7

`

1

- 7

d

• 2

c

1

2 + 11 - 22

- 3

1 + 1 - 321 - 22

`

1

- 5 + 11 - 22

d

- 2. -2R 1 + R 2.

SECTION 4.4 Solving Systems of Linear Equations by Matrix Methods 249

NOW TRY
EXERCISE 1
Use row operations to solve
the system.

2 x- 3 y=- 12

x+ 3 y= 3

NOW TRY ANSWER

1. 51 - 3, 2 26

1 in the first position of column one

0 in every position below the first

Now we go to column two. The number 1 is needed in row two, column two. We

use the second row operation, multiplying each number of row two by

1

7 R 2

c

1

0

- 3

1

`

1

- 1

d

1

7.

Stop here — this

matrix is in row

echelon form.

This augmented matrix leads to the system of equations

or

From the second equation, we substitute for yin the first equation to

find x.

Let

Multiply.

Subtract 3.

The solution set of the system is Check this solution by substitution in

both equations of the system.

51 - 2, - 126.

x=- 2

x+ 3 = 1

x- 31 - 12 = 1 y=-1.

x- 3 y= 1

y=-1, - 1

x- 3 y= 1

y=-1.

1 x- 3 y= 1

0 x+ 1 y= - 1 ,

Write the values of x

and yin the correct order. NOW TRY

NOTE If the augmented matrix of the system in Example 1is entered as matrix [A]

in a graphing calculator (FIGURE 11(a)) and the row echelon form of the matrix is

found (FIGURE 11(b)), then the system becomes the following.

While this system looks different from the one we obtained in Example 1,it is equiv-

alent, since its solution set is also 51 - 2, - 126.

y=- 1

x+

1

2

y=-

5

2

(a)

(b)

FIGURE 11