The document covers Cramer's Rule and determinants, detailing how to evaluate 2x2 and 3x3 determinants and use them to solve systems of equations. It provides examples of calculating determinants and applying Cramer's Rule to find variable values using determinants of modified matrices. Additionally, it includes classwork assignments for practice on the concepts discussed.

1. 7.8 Cramer’s Rule
Chapter 7 Systems of Equations and Inequalities

2. Concepts and Objectives
⚫ The objectives for this section are
⚫ Evaluate 22 determinants.
⚫ Use Cramer’s Rule to solve a system of equations in
two variables.
⚫ Evaluate 33 determinants.
⚫ Use Cramer’s Rule to solve a system of three
equations in three variables.
⚫ Know the properties of determinants.

3. Determinants
⚫ Every n  n matrix A is associated with a real number
called the determinant of A, written  A .
⚫ The determinant is the sum of the diagonals in one
direction minus the sum of the diagonals in the other
direction.
⚫ Example:
−3 4
6 8
= − − = −
24 24 48
( )( ) ( )( )
= − −
3 8 6 4
a b
c d
ad cb
= −

4. Determinants
⚫ Example: Find the determinant of
−
 
 
 
2 2
3 1

5. Determinants
⚫ Example: Find the determinant of
−
 
 
 
2 2
3 1
( )( ) ( )( )
−
= − −
2 2
2 1 3 2
3 1
= + =
2 6 8

6. Determinants
⚫ To calculate the determinant of a 33 matrix, repeat the
first two columns to help you draw the diagonals:
⚫ Manually calculating the determinant of a matrix larger
than 3×3 is considerably more complicated, and is really
beyond the scope of this class.
− −
−
8 2 4
7 0 3
5 1 2
−
−
−
−
−
= 7 0
5
8 2
1
4
3
8 2
7
2 5
0
1
=50
0
= ( )
30
+ − 28
+ (0
− ( )
24
+ − ( ))
28
+ −

7. Determinants (cont.)
⚫ Desmos.com/matrix calculator can also calculate the
determinant of a matrix you have entered.

8. Determinants (cont.)
⚫ Desmos.com/matrix calculator can also calculate the
determinant of a matrix you have entered.

9. Cramer’s Rule
⚫ To solve a system using Cramer’s Rule, set up a matrix of
the coefficients and calculate the determinant (D).
⚫ Then, replace the first column of the matrix with the
constants and calculate that determinant (Dx).
⚫ Continue, replacing the column of the variable with the
constants and calculating the determinant (Dy, etc.)
⚫ The value of the variable is the ratio of the variable
determinant to the original determinant.

10. Cramer’s Rule
⚫ Example: Solve the system using Cramer’s Rule.
+ = −


+ =

5 7 1
6 8 1
x y
x y

11. Cramer’s Rule
⚫ Example: Solve the system using Cramer’s Rule.
5
6 1
7 1
8
x y
x y
+ =


+ =
−

40 4
7
6 8
2 2
5
D = = − = −
7
1
1
8 7 15
8
x
D = = − − = −
−
( )
1
5
6
5 6 11
1
y
D = = − − =
−
−
= = =
−
15
7.5
2
x
D
x
D
= = = −
−
11
5.5
2
y
D
y
D

12. Classwork
⚫ College Algebra 2e
⚫ 7.8: 8-20 (4); 7.6: 16-28 (even); 7.5: 42-48 (even)
⚫ 7.8 Classwork Check
⚫ Quiz 7.6