1. Using Matrix Row Operations
Maya and her friends Amit and Nina have a lawn care business offering three services:
Lawn mowing and edging
Fertilizing and weeding
Trimming shrubs and small trees
They charge a flat rate for each service. The three partners divide up the work for a particular customer as shown below
Service
Workers and Hours
Cost
Mowing
Maya--1hr Amit--1hr Nina 1hr
$21
Fertilizing
Maya--2hr Amit--1hr
$23
Amit-- 1hr Nina--3hr
Trimming
$25

2. Using Matrix Row Operations
. Let m,a and n represent the hourly wages for Maya, Amit and Nina respectively. [ ]
m+a+n=21
2m+a=23
A+3n=25
: 21
: 23
: 25
systems
The final matrix is said to be in reduced row-echelon form.
To transform an augmented matrix into reduced row-echelon
Form, use the elementary row operations.

3. Using Row Matrix Operations
Elementary Row Operations
The following operations produce equivalent matrices, and may be used in any order and as many times as necessary to obtain reduced row-echelon form.
Interchange two rows
Multiply all entries in one row by a non-zero number.
Add a multiple of one row to another.

4. Using Matrix Row Operations
Refer to the lawn-care problem described at the beginning of the lesson.
Use the row-reduction method to solve the system.
Find the hourly wages for Maya, Amit, and Nina. Then find the total amount that each partner earns for this job. [ ]
m+a+n=21
2m+a=23
A+3n=25
: 21
: 23
: 25
systems
Perform row operations
Inspect column 1

5. Using Matrix Row Operations
The first row begins with a 1, but the 2 in the second row needs to become a 0.
Identity
Solutions [ ]
: 21
: 23
: 25 [ ]
: 8
: 7
: 6
We want to replace row 2with
The sum of row 2 and -2 times row 1
This is our goal we want our
Matrix to look like [ ]
: 21
: -19
: 25
-2R1+R R2

6. Using Matrix Row Operations[ ]
Inspect column 2 Row 1
Change it to zero
: 21
: -19
: 25
R2 + R R1 [ ]
: 2
: -19
: 25

7. Using Matrix Row Operations[ ]
Inspect column 2 Row 2
Change it to zero
: 2
: -19
: 25
-1R R2 [ ]
: 2
: 19
: 25

8. Using Matrix Row Operations[ ]
Inspect column 2 Row 3
Change it to zero
: 2
: 19
: 25
-1R2+R R2 [ ]
: 2
: 19
: 6

9. Using Matrix Row Operations[ ]
Inspect column 3 Row 1
Change it to zero
: 2
: 19
: 6
R3+R R1 [ ]
: 8
: 19
: 6

10. Using Matrix Row Operations[ ]
Inspect column 3 Row 3
Change it to zero
: 2
: 19
: 6
-2R3+R R2 [ ]
: 8
: 7
: 6
The matrix is now row echelon form
Maya=$8hr Amit=$7hr Nina=$6