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Name
Class
12-4
Date
Reteaching
Inverse Matrices and Systems
ax  by  p
a b   x 
 p
as 
  .
• You can write the system 



 cx  dy  q
c d   y
q
a b 
x
• 
is the coefficient matrix,   is the variable matrix, and

c d 
 y
 p
 q  is the constant matrix.
 
• Solve the matrix equation by multiplying both sides on the left by the inverse of the
coefficient matrix, if it exists.
4 x  3 y   4
What is the solution of the system 
? Solve using matrices.
3 x  y   3
 4 3  x 
 4 
 3 1  y    3 Write the system

 
  as a matrix
equation.
3
1
Find the inverse of

 4 3
 1 3
1  1 3
1
13
13 
 the coefficient
 3 1  det A  3 4   (1)(4)  (3)(3)  3 4    3
4







  matrix.
13
13 
1
3
3
1
1
13


13  4 3  x 
13
13   4  Isolate the
 


 



 3  4   3 1  y 
 3  4   3 variable matrix by
13
13
13 
13 
multiplying both
sides on the left
by the inverse of
the coefficient
matrix.
.
 4 
 9 
     


 x
 1
 13  
 13 
   Simplify.
 y  
 12 
 12  
 
 0
      
 13  
 13 
The solution is (1, 0). Substitute the values in the original system to check your work.
Exercises
Solve each system of equations using a matrix equation. Check your answers.
2 x  7 y   3
 x  5y  7
1. 
 x  3y  5
x  4 y  6
2. 
x  3 y  1
 5 x  16 y  5

3. 
Prentice Hall Algebra 2 • Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
39
Name
12-4
Class
Date
Reteaching (continued)
Inverse Matrices and Systems
You can use a graphing calculator and matrices to solve a linear system.
4 x  5 y  6 z  50

What is the solution of the system  x  2 y  3z  1 ? Solve using a graphing

y  z  3

Write the system as a matrix equation.
Enter the coefficient matrix as matrix A.
 4 5 6  x 
 50
1 2



3  y    1

 0 1 1  z 
 3
Enter the constant matrix as matrix B.
Multiply A1B to find the variable matrix.
The solution is ( 11.6,  21.6, 18.6). Substitute the values in the original system to check
your work.
Exercises
Solve each system of equations using a graphing calculator and matrices.
Check your answers.
 4x  y  z  0

4. 5 x  2 y  3z  15
 6 x  5 y  5 z  52

0.5 x  1.5 y  z  7

5.  3x  3 y  5 z  3
 2 x  y  2 z  1

 x  2 y – z  19

 –8
6.  2 x – 3 y

y – 5 z  16

 4 x  8 y  3z   7

 y  5z   6
7. 
2 x  5 y
 1

Prentice Hall Algebra 2 • Teaching Resources
Copyright © by Pearson Education, Inc., or its affiliates. All Rights Reserved.
40