solve the system using augmented matrices?

Question

3x+ 5y=7
6x - y= -8

Answer 1

Hi,
3x+ 5y=7
6x - y= -8

First let's convert it into an augmented matrix (augmented means that it has the constants included.)
| 3 5 7|
| 6 -1 -8|

Now, we want to cause the 3 and -1 to be 1 and the 6 and 5 to be zero, and we do that by elementary row operations, such as adding one row to another, multiplying a row by a constant, and multiplying one row by a number and adding it to another.
The really easy way to do that is to use a graphing calculator, which I’ll explain later, but let’s do it by hand.
I’ll have to use dots to maintain spaces. Ignore them, please.

| 3 5 7|
| 0 -11 -22|…….-2R1 + R2--> R2
(This is shorthand for multiply row 1 by -2, add it to row2, and replace row 2 with that result.)

| 3….5….7|
| 0….1….2|…..-1/11R2 -->R2

| 3….0….-3| -5R2 +R1--> R1
| 0….1…..2|

| 1….0….-1| 1/3R1--> R1
| 0….1…..2|

So, we read x = -1 (the one in the x-position can be read as x.)
And y = 2.

Now, how would we do this on a TI-83 Plus or TI-84. (Can’t be done this way on a TI-82 but it can be done another way.)

1) Entering a matrix:
a) Press 2nd, MATRIX, move the cursor to EDIT.
b) Move the cursor to the matrix number you want to edit or enter numbers into
and press ENTER.
c) Enter the number of rows and press ENTER; then enter the number of
columns and press ENTER.
d) Enter each value of the matrix and press ENTER after each value.
e) Press 2nd, QUIT to go to the home screen.

2) Doing rref:
a) First enter your matrix as in item 1 of this section and press 2nd, QUIT to go to the
home screen.
b) Press 2nd, MATRIX, and move the cursor to MATH.
c) Select item B for rref and press ENTER. Note that if you know you
want to use item B, just press ALPHA; then B. The expression rref( will be pasted to the
home screen.
d) Press 2nd, MATRIX and press the number for the matrix you want to operate on.
e) Press ENTER and the answer will appear.
f) If you want the answer in fractions, skip step e) and press MATH, ENTER, ENTER.
Incidentally, I copied this calculator procedure from the TI-83 Plus guide on my Website at this URL: www.anglefire.com/pro/fkizer
You’re welcome to go there and make single copies of anything you wish.

Hope this helps.
FE

Answer 2

(3 5 7)
(6 -1 -8)

R1 -> R1/3

(1 5/3 7/3)
(6 -1 -8)

R2 -> R2 -6R1

(1 5/3 7/3)
(0 -11 -22)

R2 -> R2/-11

(1 5/3 7/3)
(0 1 2)

R1 -> R1 -5/3R2

(1 0 -1)
(0 1 2)

x = -1
y = 2