4-8 Augmented
Matrices & Systems
Objectives
Solving Systems Using Cramer’s Rule
Solving Systems Using Augmented Matrices
Vocabulary
Cramer’s Rule
ax + by = m
cx + dy = n
D 
a
b
c
d
Dx 
m
b
n
d
System
Use the x- and
y-coefficients.
Then,
x
Dx
D
&
y
Dy
D
Replace the xcoefficients with
the constants
Dy 
a
m
c
n
Replace the ycoefficients with
the constants
Using Cramer’s Rule
Use Cramer’s rule to solve the system
7x – 4y = 15
.
3x + 6y = 8
Evaluate three determinants. Then find x and y.
D =
7 –4
3 6
= 54
Dx =
15 –4
= 122
8 6
Dx
61
x =
=
27
D
The solution of the system is
61 11
,
.
27 54
Dy =
y =
7
3
15
= 11
8
Dy
11
=
54
D
Using Cramer’s Rule with
Find the y-coordinate of the solution of the
–2x + 8y + 2z = –3
–6x +
2z = 1 .
–7x – 5y + z = 2
system
D =
Dy =
–2 8
–6 0
–7 –5
–2 –3
–6 1
–7 2
2
2
1
= –24
2
2 = 20
1
Evaluate the determinant.
Replace the y-coefficients with the
constants and evaluate again.
Dy
20
5
y =
=–
=–
D
24
6
Find y.
The y-coordinate of the solution is –
5
.
6
Vocabulary
An augmented matrix contains the coefficients and the constants from a system
of equations. Each row represents an equation.
System of Equations
Augmented Matrix
-6x + 2y = 10
4x
= -20
 6

 4
2
0
10 

 20 
Writing an Augmented Matrix
Write an augmented matrix to represent the
system
System of equations
–7x + 4y = –3
x + 8y = 9
–7x + 4y
x + 8y
x-coefficients
Augmented matrix
= –3
= 9
y-coefficients
–7
1
4
8
constants
–3
9
Draw a vertical bar to separate the
coefficients from constants.
Writing a System From an
Augmented Matrix
Write a system of equations for the augmented
matrix
9 –7
2 5
Augmented matrix
9
2
–1 .
–6
–7
5
x-coefficients
System of equations
–1
–6
y-coefficients
9x – 7y
2x + 5y
constants
= –1
= –6
Vocabulary
Row Operations
To solve a system of equations using an augmented matrix, you can use one
or more of the following row operations.
•Switch any two rows
•Multiply a row by a constant
•Add one row to another
•Combine one or more of these steps
The goal is to get the matrix to the left of the line into the identity
matrix. The values to the right of the line will be your solutions.
1

0
0
1
x

y
Number here will be x-value
Number here will be y-value
Using an Augmented Matrix
Use an augmented matrix to solve the system
x – 3y = –17
4x + 2y = 2
1 –3
4 2
–17
2
1 –3
–17
0 14
70
1 –3
0 1
–17
5
Write an augmented matrix.
–4(1 –3 –17)
4
2
2
0
14 70
1
14 (0
0
14
1
Multiply Row 1 by –4 and add it to Row 2.
Write the new augmented matrix.
70) Multiply Row 2 by 1 .
14
5
Write the new augmented matrix.
Continued
(continued)
1 –3
0 1
1
0
0
1
–17
5
–2
5
1
14 (0
0
14
1
70)
5
1 –3 –17
3(0 1 5)
1 0 –2
Multiply Row 2 by 3 and add it to Row 1.
Write the final augmented matrix.
The solution to the system is (–2, 5).
Check:
x – 3y = –17
4x + 2y = 2
(–2) – 3(5) –17 4(–2) + 2(5) 2
–2 – 15 –17
–8 + 10 2
–17 = –17
2=2
Use the original equations.
Substitute.
Multiply.
Using a Graphing Calculator
Use the rref feature on a graphing calculator to solve the
system
4x + 3y + z = –1
–2x – 2y + 7z = –10.
3x + y + 5z = 2
Step 1: Enter the
augmented matrix
as matrix A.
The solution is (7, –9, –2).
Step 2: Use the rref feature
of your graphing
calculator.
Continued
(continued)
Partial Check:
4x + 3y + z = –1
Use the original equation.
4(7) + 3(–9) + (–2)
–1
Substitute.
28 – 27 – 2
–1
Multiply.
–1 = –1
Simplify.
Homework
4-8 pg 224 & 225 # 1, 4, 6, 7, 9, 12, 13, 18, 19
You must do by hand and show your
work for all the steps.