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Math 141, Spring 2014, Benjamin
Aurispa
2.1 Introduction to Systems of Equations
A linear equation in 2 variables is an equation of the form ax + by = c.
A linear equation in 3 variables is an equation of the form ax + by + cz = d.
To solve a system of equations means to find the set of points that satisfy EVERY equation in the system.
A solution to a system of linear equations with 2 variables will be an ordered pair (e, f ).
A solution to a system of linear equations with 3 variables will be an ordered triple (e, f, g).
The same ideas are true for 4 variables, etc.
We can use algebraic methods to solve systems of equations, such as substitution and elimination.
Solve the following system of equations algebraically.
x − 2y = −8
3x + y = −3
We would say this system of equations has 1 unique solution.
When looking at a system of 2 linear equations, solving the system amounts to finding the points of intersection. To determine the number of solutions to a system of 2 linear equations, we can look at the slopes
and y-intercepts of each line. There are 3 cases:
• Case 1: Example from above. Solving both equations for y would yield:
y = 12 x + 4
y = −3x − 3
• Case 2: (2.1, #4, Tan):
3x − 4y = 24 ⇒ y = 34 x − 6
9x − 12y = 36 ⇒ y = 34 x − 3
• Case 3: (2.1, #8, Tan):
4x + 6y = 10 ⇒ y = − 23 x + 35
6x + 9y = 15 ⇒ y = − 23 x + 35
Summary of the three cases for the number of solutions to a system of 2 linear equations in terms of how
lines intersect.
1. Exactly one solution (a unique solution): The lines intersect at exactly one point. They have
different slopes.
2. No solution (called an inconsistent system): The lines do not intersect. They are parallel lines
with the same slope but different y-intercepts.
3. Infinitely many solutions (called a dependent system): The lines are the same line. They have the
same slope and y-intercept.
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Math 141, Spring 2014, Benjamin
Aurispa
Example: Find the value of k so that the following system of equations has no solution.
3x + ky = 12
2x − 4y = 10
Systems of equations are used to model problems. ALWAYS CLEARLY DEFINE YOUR VARIABLES
when setting up a system of equations. Set up a system of equations to solve the following problems.
Example 1: Ben has a total of $30,000 that he wants to invest in three accounts that have yields of 8%,
10%, and 11% interest per year, respectively. The amount of money invested in the 8% and 11% accounts
combined is to equal the amount of money invested at 10%. If the investment goal is to have an average
rate of return of 9.6% of the total investment, find how much Ben needs to invest in each account.
Example 2: (2.1, #24, Tan):
A theater has a seating capacity of 900 and charges $2 for children, $3 for students, and $4 for adults. At
a certain screening with full attendance, there were half as many adults as children and students combined.
The receipts totaled $2800. How many children, students, and adults attended the show?
Example 3: A person has 91 coins made up of pennies, nickels, dimes, and quarters. The total value of
the coins is $6.25 and there are 4 times as many nickels as quarters. Additionally, the value in quarters is
12 times the value in pennies. How many of each type of coin does this person have? Let p, n, d, and q be
the number of pennies, nickels, dimes, and quarters the person has respectively.
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Math 141, Spring 2014, Benjamin
Aurispa
Example 4: A furniture company makes loungers, chairs, and footstools out of wood and fabric. The
number of units of each of these materials as well as the amount of time (in minutes) needed to make each
item are given in the table below. How many of each product can be made if there are 850 units of wood,
660 units of fabric, and 19 labor-hours available and all resources are used? Just set up the problem. Let
x, y, and z represent the number of loungers, chairs, and footstools made respectively.
Wood
Fabric
Time
Lounger
40
30
45
Chair
25
20
30
Footstool
10
8
22
Example to try on your own: A sporting goods stores 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? Just set up the problem.
2.2 Unique Solutions
We can represent a system of linear equations using an augmented matrix. In general, a matrix is just a
rectangular arrays of numbers. Working with matrices allows us to not have to keep writing the variables
over and over.
To write a system of equations as an augmented matrix, line up all the variables on one side of the equal
sign with the constants on the other side. Then, form a matrix of numbers with just the coefficients and
constants from the equations.
Write the following systems using augmented matrices.
−3x + 2y − 6z = 14
4x − 5y + 2z = −3
7x + 9y + 3z = 14
2x − 4y = 9
−2y − z = 0
x + y + 4 = 2z
3y + z = 5x
If a matrix has m rows and n columns, we say the size of the matrix is m x n. What are the sizes of the
above matrices?
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Math 141, Spring 2014, Benjamin
Aurispa
In order to solve a system, we need to “reduce” the augmented matrix to a form where we can easily identify
the solution. This form is called “reduced-row echelon form.” It is equivalent to the original system, just
simplified.
A matrix is in row-reduced form if it meets the following 4 criteria:
1. Each row that has all zeros lies below any other row with nonzero entries.
2. The first nonzero entry in each row is 1, called a leading 1.
3. In any two successive nonzero rows, the leading 1 in the lower row lies further to the right of the
leading 1 in the upper row.
4. If a column contains a leading 1, then all the other entries in that column are 0.
Note: When determining if a matrix is in row-reduced form, only consider the coefficient side of the matrix
(the part to the left of the vertical line).
Examples: State whether the following matrices are in row-reduced form.


1 0 0 7


 0 4 3 2 
0 0 1 3
"
1 0
7
0 1 −9
#
"
1 −2 3
0
1 5
#


1 2 0
6


−2
0
0
1


1
0 0 0


0 1 3 8


 0 1 5 9 
0 0 1 4
"
1 0 1 4
0 1 3 7
#
Once a matrix is in row-reduced form, the solution can be found by rewriting each row as an equation.
What is the solution to the system represented by the reduced matrix below?


1 0 0 8


 0 1 0 1 
0 0 1 3
There are 3 row operations that can be used to get a matrix in row-reduced form:
1. Interchange any two rows. (This one is not used very often.)
Notation: Ri ↔ Rj means to interchange row i with row j.
2. Replace any row by a nonzero constant multiple of itself.
Notation: cRi → Ri means to replace row i with c times row i.
3. Replace any row by the sum of that row and a constant multiple of another row.
Notation: Ri + aRj → Ri means to replace row i with the sum of row i and a times row j.
Example: Perform the given row operations in succession on the matrix.


R2 − 3R1 → R2
R3 + 2R1 → R3
1 3
1 3


3 7 
 3 8
−2 3 −1 10
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Math 141, Spring 2014, Benjamin
Aurispa
If an augmented matrix for a system of equations is not in row-reduced form, we use the Gauss-Jordan
Elimination Method to get it reduced.
1. Write the system as an augmented matrix.
2. Look at the first entry in the first row. Make this entry into a 1 and all other entries in that column 0s.
This is called “pivoting” the matrix about this element. (If the entry is a 0, you must first interchange
that row with a row below it that has a nonzero first entry.)
3. Once this is done, move down the diagonal to the second entry of the second row and repeat the
process.
4. Continue until the whole matrix is in row-reduced form.
Solve the following system of equations using the Gauss-Jordan method:
−6x − 12y = 6
−4x − 5y = 10
Example: (2.2 #19, Tan) Pivot the matrix about the boxed element.
5
"
2 4 8
3 1 2
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Math 141, Spring 2014, Benjamin
Aurispa


1
3 −1 −3


4 .
What would be the next two steps in the Gauss-Jordan method for the matrix  0
1 −2
7
0 −4 −1
The calculator command rref will reduce the matrix for you without having to do algebra.
To enter a matrix into your calculator and find the row-reduced form:
• Press 2nd x−1 . (The TI-83 has a separate MATRX button.)
• Cursor over to EDIT and select where you want to store your matrix. Enter the size of the matrix
and then enter in the matrix values.
• Go back to your home screen. Press MATRX and then cursor right to MATH. Scroll down and select
B:rref.
• Select the matrix you want to reduce by pressing MATRX and then choosing your matrix under the
NAMES column. Then close your parentheses and press enter. Note: The rref command only works
when the number of rows is less than or equal to the number of columns.
Now, we’ll finish some of the problems we set up earlier.
Example 1:
x + y + z = 30000
.08x + .10y + .11z = 2880
x+z =y
Example 3:
p + n + d + q = 91
.01p + .05n + .1d + .25q = 6.25
n = 4q
.25q = 12(.01p)
2.3 Systems with No Solution or Infinitely Many Solutions
We have already seen some examples of systems of equations with infinitely many solutions or with no
solutions. When the systems get larger, it is much easier to use the calculator.
Examples: Find the solutions represented by the following row-reduced matrices. Assume that the variables
these matrices represent are x, y, and z.





1
0
0
0
0
1
0
0
0
0
1
0
2
1
3
1





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Math 141, Spring 2014, Benjamin
Aurispa


1 0 −3
4


2 −1 
 0 1
0
0 0
0
When a system has infinitely many solutions, we parameterize the solution (write the solution in parametric
form):
A particular solution is a specific solution to the system. What are some particular solutions for this system?
Solve the following system of equations:
x + 2y + 3z = 2
−3x + 3y = −1
7x − 13y + 7z = −1
−x + 7y + 4z = 3
Examples: The following reduced matrices represent systems of equations with variables x, y, z, and, if
necessary, u. Determine the solutions to these systems.
#
"
1 5 0 3
0 0 1 0

1
0
0
0
0
1
0
0
0
0
1
0
3
0
0 −4 


2
3 
1
0

1
0
0
0
0
1
0
0
2
5
0
0
0 −5
0
0 


2 
1
0
0










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