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Math 166, Fall 2015, Robert
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4.3- Systems of Linear Equations
An equation is called linear if every variable present is raised to the first power
and variables are not multiplied together. For example, every linear equation
in two variables, x and y, can be written in the form
ax + by = c
for some numbers a,b, and c. To solve a system of linear equations, we need to
find the values of the variables that solve every equation simultaneously.
There are only three possible solution sets to systems of linear equations:
1. Exactly one solution (called the unique solution to the system)
2. No solution (in which case we say the system of equations is inconsistent)
3. Infinitely many solutions (in which case we say that the system is dependent)
When working a word problem, always clearly define your variables.
Example 1 Solve the following system of linear equations:
x + y + z = 14
4y − 5z = −10
y + 3z = 23
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Math 166, Fall 2015, Robert
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Example 2 A restaurant is famous for two dishes. When they sell the first
dish, the restaurant makes a profit of $12. When they sell the second, they make
a profit of $8. If the restaurant makes a profit of $800 dollars after selling a
combined total of 75 of these dishes, how many of each were sold?
We can encode all of the important information about a linear system of equations into a rectangular array of numbers called a matrix. The matrix will keep
track of all coefficients and constants without the need to constantly rewrite every variable. We often use augmented matrices, meaning that we use a line
to separate the entries from our coefficients from the solutions of the individual
equations. The equations from Example 1 were

x + y + z = 14
1 1 1


4y − 5z = −10 as a matrix this is 0 4 −5

y + 3z = 23
0 1 3
14



−10

23
Example 3 Encode the following system of equations into an augmented matrix
3x + 2y − 4z + w = 3
2x − 3z + 6 = 4w
z + x − y = 14
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Math 166, Fall 2015, Robert
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We may “solve” an augmented matrix by using Gaussian Elimination or GaussJordan Elimination. The goal of Gaussian Elimination is to reduce a matrix
into row echelon form. This means that
• all nonzero rows are above any zero rows
• the first nonzero number in any row is a 1
• every number below this leading one is a 0
Gauss-Jordan Elimination takes Gaussian Elimination a step further by turning
a matrix into reduced row echelon form. This is the same as row echelon form
but with the following extra condtion: the leading 1 of each row is the only
nonzero number in its column.
Example 4 Which of these matrices are in row echelon form? Reduced row
echelon form?


1 3 5




0 1 2


0 0 0

1


0

0
3
0
1

5


0

2

2


0

0
1
3
0

5


4

1

1


0

0
0
1
0

4


3

0
We get matrices into this form by using the following elementary row operations:
• Interchange the ith row with the j th row (Ri ↔ Rj )
• Multiply each member of the ith row by k, a nonzero constant (kRi → Ri )
• Replace each element of the ith row with the corresponding element in the
ith row plus k times the corresponding element in the j th row
(Ri + kRj → Ri )
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Math 166, Fall 2015, Robert
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Example 5 Solve the following system of equations
x + 2y − z = 1
3x + 5y + z = 10
x + 3y + z = 6
Example 6 A local mechanic specizlizes in repairing cars and motorcycles. A
typical job on a car takes him 4 hours to complete, while a typical job on a
motorcycle takes him 2 hours to complete. If the mechanic works for 40 hours
and wants to repair twice as many cars as he does motorcycles, how many of
each should he repair?
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Math 166, Fall 2015, Robert
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4.4- Systems with Non-Unique Solutions
Your calculator has a function that will reduce matrices to reduce row echelon
form without you having to do any algebra:
• Press 2nd x−1 (Some calculators may have a separate MATRX button)
• Move to EDIT and select a place to store your matrix. Enter the size of
the matrix, then its values.
• Return to the MATRX menu and then move the cursor over to MATH.
Scroll down to find rref
• Select the matrix that you want to reduce
Note that this only works when the number of rows in your matrix is less than or
equal to the number of columns. When a system has infinitely many solutions,
we parameterize the solution- that is we write all solutions in terms of a new
variable
Example 7 For the following matrix, write the corresponding system of equations and solve the system.

1 0 6 0

0 1 0 8

5

7
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Math 166, Fall 2015, Robert
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Example 8 For the following matrix, write the corresponding system of equations and solve the system.

1


0

0
0
3
1
0
0
0

2


8

1
Example 9 Solve the following system of equations
2x + 3y + z = 15
−x + y + 4z = 7
4x + 6y + 2z = 30
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Math 166, Fall 2015, Robert
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Example 10 Amy has a total of $5000 invested in two different stocks. Over
the last year, one of the stocks returned a 4% profit on the money invested, while
the other returned a 7% profit on the money invested. If Amy made $1404.94
profit that year, how much money did she have invested in each stock?
Example 11 A movie theater sells tickets to adults for $9.25, to children for
$5, and to students for $7.50. On a particular night, the theater sells 568 tickets
for a total of 4383.75. If five times as many adult tickets were sold as student
tickets, how many of each type of ticket was sold?
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Math 166, Fall 2015, Robert
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Example 12 A team of artists decide that they should work together over 3
days to create new paintings and sculptures. It takes the team 5 hours to create
a new sculpture and 3 hours to create a new painting. Based on their previous
works, they project that they should be able to sell each sculpture for $175 and
each painting for $105. If the team decides to spend a total of 23 hours working
on their art and want to make $805 from selling it, how many of pieces of each
type should they make?
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