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Math 141-copyright Joe Kahlig, 10B
Section 2.2: Systems of Linear Equations: Unique Solutions
Section 2.3: Systems of Linear Equations: Underdetermined and Overdetermined Systems
Solving System of Equations.
Definition: An augmented matrix is a condensed method of representing a system of equations.
Example: Represent the system of equations as an augmented matrix.
3x + 2y = 7
x + 4y = 10
Example: Give the system of equations represented by the augmented matrix. (Variables are listed
in the first row.)

x

 1
3
y
-4
4

z
2
2
10 
20

Row Operations are used to manipulate an augmented matrix into a form (usually row reduced
form) where the solution can easily be discerned. The three row operations are:
1) Swap Rows
2) Multiply a row by a non-zero number
3) Add a multiple of one row to another row.
Row Reduced Form (reduced row echelon form)
1. The first non-zero number in a row is a 1(called a leading one).
2. The leading one is the only non-zero number in a column.
3. The leading ones are in a diagonal like fashion from the upper left to the lower right.
Example: Which of these matrices are in row reduced form?

1

A)  0
0
C)
"
1
0
0
1
0
3
0

0
0
1
3

8 
2
0
1
#
7
6

1

B)  0
0
0
1
0
3
0
2
5

6 
7

0
1
0
0
0
0
3

7 
10
1

D)  0
0


Math 141-copyright Joe Kahlig, 10B
Example: Solve the system of equations.
3x + 2y = 7
x + 4y = 10
Example: Create a leading one in the row one column one position.
3x + y + 2z = 11
4x + 9y + z = 25
2x − y + 3z = 9
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Math 141-copyright Joe Kahlig, 10B
Example: Solve these system of equations.
A)
3x + y − 9 = 0
x−y+z−4=0
3x + z − 11 = 0
4x − y + 2z = 15
B)
x + y − 3z = 0
2x − 3y + z = 1
4x − y − 5z = 1
C)
x + 3y − z − 3w = 7
2x + 4y − 2w = 10
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Math 141-copyright Joe Kahlig, 10B
Example: The figure shows the flow of traffic during rush hour on a typical weekday. The arrows
indicate the direction of traffic flow on each one–way road, and the average number of vehicles entering
and leaving each intersection per hour appears beside each road. Fifth and Sixth Avenues can handle
up to 2000 vehicles per hour without causing congestion, whereas the maximum capacity of each of
the two streets is 1000 vehicles per hour. The flow of traffic is controlled by traffic lights installed at
each of the four intersections.
5th st.
4th st.
1. Set up the system of equations that
would model this problem.
5th Av.
1500
2. Solve the system of equations and write
the answer in parametric form. Place
restrictions on the parameter.
3. Find two possible flow patters that
would ensure that there is no traffic congestion.
350
500
x
1200
y
w
z
1400
6th Av.
1100
550
400
Math 141-copyright Joe Kahlig, 10B
Example: Give the solution for this problem.
x = the number of small drinks
y = the number of medium drinks
z = the number of large drinks





x
1
0
0
y
0
1
0
z
-0.5
4
0
-3
59
0





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