# Page 1 Section 4.3: Gauss Elimination for System of Linear Equations

```Math 166-copyright Joe Kahlig, 10A
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Section 4.3: Gauss Elimination for System of Linear Equations
Section 4.4: System of Linear Equations with Non-Unique Solutions
Example: Set up the following word problems. Do not solve (yet). Be sure that the variables are
defined.
1. You own a hamburger stand and your current inventory includes 86 bread rolls, 100 beef patties, and 140 cheese slices. Your menu consists of three types of hamburgers: plain, double
cheeseburger, and regular cheeseburger. Each plain hamburger require 1 beef patty and 1 bread
roll. Each double cheeseburger requires 1 bread roll, 2 beef patties, and 4 cheese slices. Each
regular cheese burger requires 2 cheese slices, 1 bread roll, and 1 beef patty. How many of each
hamburger should you make so that all resources are used?
2. The city of Aurora gets a matching grant of \$1 million ( making a total of \$2 million) to spend
in these categories: streets, sewers, and parks. The city council decides that the amount spent
on parks should equal the total amount spent on streets and sewers and that, furthermore, twice
as much should be spent on parks as on sewers. How much money is to allotted to each of these
categories?
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3. Link has \$7,000 to invest. He decides to invest in three different companies. The QX company
costs \$25 per share and pays dividends of \$0.75 per share each year. The RY company costs \$40
per share and pay dividends of \$1.50 per share each year. The KZ company costs \$15 per share
and pays \$2.00 per share per year in dividends. Link wants to have twice as much money in the
RY company as in the KZ company. Link also wants to earn \$352 in dividends per year. How
much should Link invest in each company to meet his goals?
4. A brokerage firm packaged blocks of blue-chip stocks, bonds and high-risk stocks into three
portfolios, which it offers to its customers. The makeup of each portfolio is given in the table. A
customer wants eight blocks of blue-chip stocks, 11 blocks of bonds, and nine blocks of high-risk
stocks. How many of each portfolio should the customer purchase?
Portfolio
I
II
III
blue chip stocks
1 block
2 blocks
4 blocks
bonds
4 blocks
1 block
2 blocks
high-risk stocks
3 blocks
2 blocks
1 block
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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)
2)
3)
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)
&quot;
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


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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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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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.
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
1500
5th Av.
x
1200
y
w
z
1400
6th Av.
1100
550
400
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
-1
4
0
-5
47
0





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