Chapter 4: Systems of Linear Equations and Models
Section 4.3-4.4
4.3 Gauss Elimination for Systems of Linear Equations
A linear equation in 2 variables is an equation of the form ax + by = c.
A solution to a linear equation in 2 variables is an ordered pair (x, y) that satisfies the
equation.
If we graph the set of all solutions to a linear equation, it will form a line.
y
x
A linear equation in 3 variables is an equation of the form ax + by + cz = d. A solution
to a linear equation in 3 variables is an ordered pair (x, y, z) that satisfies the equation.
The same ideas are true for 4 variables, 5 variables, etc.
We will work with systems of linear equations. A system of equations is a group of more
than one equation. To solve a system of equations means to find the set of points that
satisfy EVERY equation in the system.
We can use algebraic methods to solve systems of equations, such as substitution and
elimination.
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Given a system of linear equations, one and only one of the following may
occur:
1. The system has a unique solution.
y
x
Example 1 Find all solutions to the following system of equations, whenever they
exist.
2x − 4y = 5
3x + 2y = 6
2
2. The system has infinitely many solutions.
y
x
Such a system is said to be
.
Example 2 Find all solutions to the following system of equations, whenever they
exist.
5x − 6y = 8
10x − 12y = 16
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3. The system has no solutions.
y
x
Such a system is said to be
.
Example 3 Find all solutions to the following system of equations, whenever they
exist.
−10x + 15y = −3
4x − 6y = −3
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Instead of using algebra, we will learn a way to solve systems of equations using matrices
and the calculator.
is a rectangular array of numbers.
A
We can represent a system of equations with an
matrix. An
augmented matrix is a short-hand way of representing a system without having to write
the variables.
When writing an augmented matrix from a system of equations, line up all the variables
on one side of the equal sign with the constants on the other side. Then, form a matrix
with the coefficients and constants in the equations.
Example 4 Write the following system of equations as an augmented matrix.
3x + 4y + 5z = 2
2x − 2y = −1
− x + 5y + z = 3
Example 5 Write the following system of equations as an augmented matrix.
2x − 4y = 9
− 2y − z = 0
x + y + 4 = 2z
3y + z = 5x
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Example 6 Write the system of equations that correspond to the following augmented
matrix.


2 1 0
9
 4 3 6
3 
3 7 −1 −2
If a matrix has m rows and n columns, we say the size of the matrix is m × n. What are
the sizes of the above matrices?
In order to solve a system, we need to “reduce” the matrix to a form where we can
identify the solution. This form is called
.
It is equivalent to the original system, just simplified.
Reducing Matrices using Your Calculator:
1) Enter the matrix: Press 2nd x−1 . Scroll to the right to EDIT. Scroll down to your
desired matrix (say, 1:[A]). Press ENTER . Enter the dimensions of your matrix,
pressing ENTER after each dimension. Enter each entry reading left to right and
top to bottom, pressing ENTER after each entry. Press 2nd MODE to return to
the home screen.
2) Press 2nd x−1 . Scroll to MATH.
3) Select rref(. Press ENTER .
4) Press 2nd x−1 . Select your desired matrix ([A]).
5) Press ).
6) Press ENTER .
Once you have the reduced form of the matrix, you can determine the solution by rewriting
each row back as an equation.
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Example 7 Solve the following systems of equations.
a)
2x − y − z = 0
3x + 2y + z = 7
x + 2y + 2z = 5
b)
3x − 9y + 6z = −12
x − 3y + 2z = −4
2x − 6y + 4z = 8
c)
3x − 4y = 10
− 5x + 8y = −17
3x + 12y = −12
Note: In general, if you have more equations than variables, any solution is possible
(unique, no solutions, or infinitely many solutions.)
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d)
−x + 2y + 3z = 14
2x − y + 2z = 2
−x + 5y + 11z = 44
When a system has infinitely many solutions, we
the
):
solution (we write the solution in
A particular solution is a specific solution to the system. What are some particular
solutions for this system?
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Example 8 Solve the following system of equations.
7x + 2y − 2z − 4w + 3v = 8
−3x − 3y + 2w + v = −1
4x − y − 8z + 20v = 1
Note: In general, if you have more variables than equations, then you will either have
no solution or infinitely many solutions.
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Example 9 The following reduced matrices represent systems of equations with variables
x,y,z, and if necessary, u. Determine the solutions to these systems.

0 3
0 −4 

2 3 
0 1

1
 0
a) 
 0
0
0
1
0
0
0
0
1
0

1
 0
b) 
 0
0
0
1
0
0

0 6
0 −9 

1 0 
0 0

0
1
0
0
2
5
0
0
1
 0
c) 
 0
0
d)

0 −5
0 0 

1 2 
0 0
1 0 2 −1 6
0 1 −3 4 1
10
e)
1 0 0 0
0 0 1 3
Systems of equations are used to model real-world problems. For the following examples,
set up a system of equations to solve the problem and solve. ALWAYS DEFINE YOUR
VARIABLES when setting up a system of equations.
Example 10 You have a total of $4000 on deposit with two savings institutions. Institution A pays simple interest at the rate of 3% per year, whereas Institution B pays simple
interest at the rate of 4% per year. If you earn a total of $125 in interest during a single
year, how much do you have on deposit in each institution?
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Example 11 A sporting goods store sells footballs, basketballs, and volleyballs. A football
costs $35, a basketball costs $25, and a volleyball costs $15. On a given day, the store
sold 5 times as many footballs as volleyballs. They brought in a total of $3750 that day,
and the money made from basketballs alone was 4 times the money made from volleyballs
alone. How many footballs, basketballs, and volleyballs were sold?
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Example 12 (From Tan #26 2.3) A dietitian wishes to plan a meal around three foods.
The meal is to include 880 units of vitamin A, 3380 units of vitamin C and 1020 units
of calcium. The number of units of the vitamins and calcium in each ounce of the foods
is summarized in the following table:
Vitamin A
Vitamin C
Calcium
Food I
400
110
90
Food II
1200
570
30
Food III
800
340
60
Determine the amount of each food the dietitian should include in the meal in order to
meet the vitamin and calcium requirements. Just set up the problem.
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Example 13 (Tomastik/Epstein #40, 4.3) A furniture company makes loungers, chairs,
and footstools out of wood, fabric, and stuffing. The number of units of each of these
materials needed for each of the products is given in the table below. How many of each
product can be made if there are 1110 units of wood, 880 units of fabric, and 660 units
of stuffing available? Just set up the problem. Let x,y, and z represent the number of
loungers, chairs, and footstools made, repectively.
Wood Fabric
Lounger
40
40
Chair
30
20
Footstool
20
10
14
Stuffing
20
20
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If a system of equations arising from a word problem has a parametric solution, sometimes
there really aren’t infinitely many solutions, since only some of them will make sense in
the problem.
For example, if x represents the number of children that were at a movie, we would know
that x ≥ 0 and that x must be a whole number (an integer).
Example 14 (Problem #49, Section 4.4) A person has 36 coins made up of nickels,
dimes, and quarters. If the total value of the coins is $4. How many of each type of coin
does this person have? Let x, y, and z be the number of nickels, dimes, and quarters the
person has respectively.
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Example 15 An instructor wants to write a quiz with 9 questions where each question
is worth 3, 4, or 5 points based on difficulty. He wants the number of 3-point questions
to be 1 more than the number of 5-point questions, and he wants the quiz to be worth a
total of 35 points. How many 3, 4, and 5 point questions could there be? Let x, y, and z
be the number of 3-point, 4-point, and 5-point questions respectively.
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