Algebra 2
Chapter 3: Systems of Equations
Mrs. Leahy
3.1 Solving Systems Using Tables and Graphs
A solution to a system of linear equations is an _______________________________ that makes all of
the equations ___________.
To solve a system of equations by graphing:
1. ______________ each line very carefully.
2. _____________ for where the lines intersect.
3. _____________ the solution in both equations.
Example: Solve each system by graphing
1.
 y  2x 1


1
 y   2 x  4
 




2.




1

 y   4 x  3

 y   3 x 1

4



























    

































1



Algebra 2
Chapter 3: Systems of Equations
8 x  4 y  12  y  2 x  3


2 x  y  3
 y  2 x  3
3.
Mrs. Leahy
7 x  y  6
 y  7x  6


7 x  y  6  y  7 x  6
4.


















 




















 


































So what words do we use to describe these systems?
Consistent:
has at least ______ solution
Independent: has _________________ solution
Inconsistent:
has _____ solution
Dependent: has __________________ solutions.
2



Algebra 2
Chapter 3
Mrs. Leahy
3.2 Solving Systems Algebraically
Sometimes, it is difficult to graph equations accurately by hand.
Sometimes, the points of intersection are not integer values and are difficult to see.
Sometime, we need a different way to find the solution to a system of equations.
A: Solving Systems by Substitution
Example: Solve by substitution
 x  2 y  5
1.

x  2  3y
Make a Plan:
This method works well when
 One equation is already solved
for one variable in terms of
another like y= or x =

One equation has a term with
a coefficient of 1 or -1.
Make it Happen:
2.

If necessary, solve one
equation for one of the
variables.
Get x = or y =

Substitute the expression for
the variable in the other
equation.

Solve the equation. You will
almost always have to
distribute first.

Substitute your solution into
either original equation.

Solve for the remaining
variable.

Write your answer as an
ordered pair.

CHECK YOUR ANSWER IN BOTH
EQUATIONS!
 y  2x  6

7 x  6 y  1
3
Algebra 2
Chapter 3
Mrs. Leahy
(Solve by Substitution, Continued)
3.
4 x  3 y  10

 x  2 y  10
 y  2x 1

5. 
1
 y   2 x  4
4.
2 x  y  12

3x  7 y  1
3x  2 y  14
6. 
2 x  2 y  6
4
Algebra 2
B: Solving by Elimination
7.
3x  2 y  14

2 x  2 y  6
Chapter 3
Mrs. Leahy
Make a Plan:
This method works well when
 No variable is already solved for

No coefficient is 1 or -1
Make it Happen:
8.
9.
4 x  3 y  24

4 x  7 y  36

GOAL – to create additive opposites
Ex. 4x, -4x or -7y, 7y

If necessary, multiply one or both
equations by a constant to create
additive opposites.

Add the two equations. This should
eliminate one of the variables.

Solve for the variable.

Substitute into either original
equations and solve for the
remaining variable.

Write your answer as an ordered
pair.

CHECK YOUR ANSWER IN BOTH
EQUATIONS!
 x  2 y  10

3x  y  9
5
Algebra 2
10.
2 x  7 y  4

3x  5 y  5
11.
4 x  3 y  18

7 x  2 y  25
12.
 x  y  2

2 x  2 y  0
13.
4 x  y  6

12 x  3 y  18
Chapter 3
Mrs. Leahy
6
Algebra 2
Chapter 3
Mrs. Leahy
3.5 Systems with Three Variables
In lesson 3.1 & 3.2, we learned about systems of 2 equations with 2 variables, that may have a solution
(x, y) that makes both equations true.
In this lesson, we will learn about systems of 3 equations and 3 variables that may have a solution
(x, y, z) that makes all three equations true.
Example: Solve each system of equations. Check your answers.
Make a Plan

GOAL: Use elimination (lesson 3.2)
to change your 3 x 3 system to a 2 x
2 system.

Look for a variable that can easily
be eliminated from all three
equations.
1.
 x  y  z  1

 x  y  3z  3
2 x  y  2 z  0

Make it Happen

Choose a variable to eliminate.

Choose 2 equations & eliminate the
variable.

Choose a different pair of equations
& eliminate the SAME VARIABLE
again.

You now have TWO new equations
each with TWO variables. Use
substitution or elimination to solve
(Like 3.2)

Substitute your solutions into one
of the ORIGINAL 3 variable
equations and solve for the missing
term.

Write your answer as an
alphabetical ordered triplet. (x, y, z)
7
Algebra 2
2.
 x  y  2 z  7

3x  y  2 z  7
 x  3 y  z  9

3.
 x  2 y  3z  12

2 x  y  2 z  5
2 x  2 y  z  4

Chapter 3
Mrs. Leahy
8
Algebra 2
Chapter 3
Mrs. Leahy
3.6 Using Matrices to Solve Systems of Equations
We will be using a graphing calculator to do the major computation in this section. On the test, you will
be require to solve problems with the graphing calculator on your own, so be sure to pay close attention
today!
About Matrices
A matrix is a rectangular array of numbers, symbols, or expressions.
The individual items in the matrix are called “elements” or “entries.”
The matrix at the right has _____ elements.
 2 0 3
11 4 9 


The size of a matrix is given by the number of rows 1st and then the number of columns second.
The matrix at the right has _____ rows and _____ columns.
We say the size of this matrix is: _____________.
The elements in a matrix are named
Name the elements:
ai , j
a1,3 = _____
where i = row position and j = column position.
a2,1 = _____
Augmented Matrices
An augmented matrix can be used to represent a system of equations.
It is important that each equation in the system be in standard form ( Ax  By  ...  D ), that is, all the variable
terms are on the left side of the equals sign in alphabetic order and the constant term isolated on the right side of
the equals sign.
System of Equations in Standard Form
Augmented matrix
5 x  2 y  z  5

3x  3 y  3z  9
x  2 y  4z  6

5 2 1 5 
3 3 3 9 


1 2 4 6
Write the following system of equations as augmented matrices:
x  y  z  6

1. 2 x  3 y  2 z  2
3x  5 y  4 z  4

 x  2 y  10
2 x  3 y  1
2. 
9
Algebra 2
Chapter 3
Mrs. Leahy
Reduced Row Echelon Form
We are going to use a graphing calculator to change the augmented matrix for a system of equations into the
reduced row echelon form (RREF) of the matrix. From RREF, we will be able to read the solutions of the system of
equations that it represents.
Example of RREF:
Solutions:
1 0 0 5 
0 1 0 3


0 0 1 2 
Read the solutions off the following RREF matrices:
1.
1 0 0 3 
0 1 0 1


0 0 1 0 
2.
1 0 17 
0 1 13.5


TI-83 Matrix Menu
About the Matrix (MATRX) menu:
MATRX
2nd
x-1

NAMES – used to paste the name
of a matrix into the home screen
or into a program.

MATH – contains all the
operations that can be done with
a matrix.

EDIT – where you set the size of
the matrix and enter the
elements.
*NOTE – the TI 83 “may” have a Matrix button instead of a Matrix you have to shift to use. This is actually easier,
because it is one less step to make.
10
Algebra 2
Chapter 3
Mrs. Leahy
Example: SOLVE the following system
5 x  2 y  z  5

3x  3 y  3z  9
x  2 y  4z  6

INSTRUCTIONS
1. Write your system as an augmented matrix and determine the size.
2. Go to MATRX and EDIT.
ENTER to select [A]
3. Define [A] to be a 3 x 4 matrix
Use the cursor and ENTER to move through the spaces.
4. Enter the coefficients and constants from the system (your augmented matrix)
by entering the value and pressing ENTER.
5. Return to the home screen
2nd MODE to “QUIT”
6. Go to MATRX and MATH.
Scroll down and hit ENTER to select B: rref(
7. Go to MATRX and NAMES. Hit ENTER to paste [A] to your home screen.
Close the parenthesis.
8. Press ENTER and read the solution from the matrix.
SOLUTION:
x = ____
y = _____
z = _____
11
Algebra 2
Chapter 3
Mrs. Leahy
Use a graphing calculator to solve the following systems of equations.
1.
1.5 x  3.25 y  7
3 x  7.5 y  0
3.
2 x  y  3z  15

3x  4 y  2 z  7
2 x  2 y  5 z  22

2.
2 x  5 y  11

 x  y  2
4.
2 x  4 y  3z  3

x  2 y  7z  5
3x  5 y  3

Day 2: Set up a system of equations and solve using a graphing calculator.
Does your answer make sense?
1.
The tickets for the school play are $7.00 for adults and $5.00 for students.
There were 100 tickets sold. If the total profit was $550.00, how many of each were sold?
12
Algebra 2
Chapter 3
Mrs. Leahy
2.
You work at a fruit stand that sells apples for $2 per pound, oranges for $5 per pound, and bananas for $3
per pound. Yesterday you sold 60 pounds of fruit and made $180. You sold 10 more pounds of apples
than bananas. How many pounds of each kind of fruit did you sell?
3.
Suppose you want to fill nine 1-1b tins with a snack mix. You have $15 and plan to buy almonds for $2.45
per pound, hazelnuts for $1.85 per pound, and raisins for $0.80 per pound. You want the mix to contain
an equal amount of almonds and hazelnuts and twice as much of the nuts as raisin by weight. How many
of each ingredient should you buy?
4.
A hardware store mixes paints in a ratio of two quarts red to six quarts yellow to make two gallons of
pumpkin orange. A ratio of five quarts red to three quarts yellow makes two gallons of pepper red. A
gallon of pumpkin orange sells for $25 and a gallon of pepper red sells for $28. Find the cost of a quart of
red and a quart of yellow.
13