CHAPTER 4: MATRICES
Section 4.6: Row Operations and Augmented Matrices
TARGET:
•
I can use elementary row operations to solve systems
of equations.
WARM-UP:
1. Solve.
a.
"3x + y = 15
#
$3x ! 2y = 6
! x + 2y = 18
b. "
#3x + 4y = 44
2. What are the three types of linear systems?
Methods we have already used to solve systems of equations:
• Cramer’s Rule
• Inverse Matrices
For solving large systems, a different method using an augmented matrix is required.
Augmented Matrix: Consists of the coefficients and constant terms of a system of linear
equations.
A vertical line separates the
coefficients from the constants.
REPRESENTING SYSTEMS AS MATRICES
2. Write the augmented matrix for the system of equations.
"6x ! 5y = 14
a. #
$2x +11y = 57
Step 1: Write each equation in
the form ax + by = c
6x ! 5y = 14
2x +11y = 57
Gibson 2011-2012
Step 2: Write the augmented matrix, with
coefficients and constants.
"6 ! 5 14 %
$
'
#2 11 57 &
4.5 Notesheet – page 1
! x + 2y = 12
#
b. "2x + y + z = 14
# y + 3z = 16
$
Step 1: Write each equation in
the form Ax + By + Cz = D
Step 2: Write the augmented matrix, with
coefficients and constants.
"!5x !12 = 4y
$
c. #z = 3 ! x
$10 = 3z + 4y
%
Step 1: Write each equation in
the form Ax + By + Cz = D
Step 2: Write the augmented matrix, with
coefficients and constants.
SOLVING SYSTEMS WITH AN AUGMENTED MATRIX:
To solve a system using an augmented matrix:
• First you will do a row operation to change
the form of the matrix.
(These row operations create a matrix
equivalent to the original matrix; new matrix
represents a system equivalent to the original
system.)
Row reduction: the process of performing elementary row operations on an augmented matrix
to solve a system. Goal: to get the coefficients to reduce to the identity matrix on the left side.
This is called reduced row-echelon form.
3. Write the augmented matrix and solve:
"2x + y = 11
a. #
$3x ! 2y = 6
Gibson 2011-2012
4.5 Notesheet – page 2
"8x ! 5y = 18
b. #
$5x + 8y = !11
! 4x + 4y = 32
c. "
# x + 3y = 16
"3y = 15 ! 9x
d. #
$!6x = 2y +10
4. Solve by using row reduction on a calculator.
a. A shelter receives a shipment of items worth $1040. Bags of cat food are valued at $5
each, flea collars at $6 each, and catnip toys at $2 each. There are 4 times as many
bags of food as collars. The number of collars and toys together equals 100. Write the
augmented matrix and solve, using row reduction, on a calculator. How many of each
item is in the shipment?
Use the facts to write three equations. (Then use calculator to enter the 3x4 matrix as A)
Press
, select MATH, and move down the list to B:rref( to find the reduced
row-echelon form of the augmented matrix.
Gibson 2011-2012
4.5 Notesheet – page 3
"3x ! y + 5z = !1
$
b. # x + 2z = 1
$ x + 3y ! z = 25
%
c. A new freezer costs $500 plus $0.20 a day to operate. An old freezer costs $20 plus
$0.50 a day to operate. After how many days is the cost of operating each freezer
equal? Let t represent the total cost of operating a freezer for d days.
Gibson 2011-2012
4.5 Notesheet – page 4