1. Solving Linear Systems Using Inverses and Cramer’s Rule
LESSON 6–3
Solving Linear Systems Using Inverses and Cramer’s Rule

2. Five-Minute Check (over Lesson 6-2) Then/Now New Vocabulary
Key Concept: Invertible Square Linear Systems
Example 1: Solve a 2 × 2 System Using an Inverse Matrix
Example 2: Real-World Example: Solve a 3 × 3 System Using an Inverse Matrix
Key Concept: Cramer’s Rule
Example 3: Use Cramer’s Rule to Solve a 2 × 2 System
Example 4: Use Cramer’s Rule to Solve a 3 × 3 System
Lesson Menu

3. Find AB and BA, if possible.
A. BA is not possible; AB =
B. AB is not possible; BA =
C. AB = ; BA =
D. AB is not possible; BA is not possible
5–Minute Check 1

4. Write the system of equations as a matrix equation, AX = B
Write the system of equations as a matrix equation, AX = B. Then use Gauss-Jordan elimination on the augmented matrix to solve for X. x1 + 2x2 + 3x3 = –5 2x1 + x2 + x3 = 1 x1 + x2 – x3 = 8
5–Minute Check 2

5. A. ; (1, 3, –4) B. ; (–1, –3, 4) C. ; (1, 3, –4) D. ; (–1, –3, 4)
5–Minute Check 2

6. For and , find AB and BA and determine whether A and B are inverse matrices.
5–Minute Check 3

7. Which of the following represents the determinant of ?
B. 13
C. 15
D. 17
5–Minute Check 4

8. Solve systems of linear equations using inverse matrices.
You found determinants and inverses of 2 × 2 and 3 × 3 matrices. (Lesson 6-2)
Solve systems of linear equations using inverse matrices.
Solve systems of linear equations using Cramer’s Rule.
Then/Now

9. square system
Cramer’s Rule
Vocabulary

10. Key Concept 1

11. Write the system in matrix form AX = B.
Solve a 2 × 2 System Using an Inverse Matrix
A. Use an inverse matrix to solve the system of equations, if possible. 2x – y = 1 2x + 3y = 13
Write the system in matrix form AX = B.
AX = B.
Example 1

12. Formula for the inverse of a 2 × 2 matrix. A–1
Solve a 2 × 2 System Using an Inverse Matrix
Use the formula for the inverse of a 2 × 2 matrix to find the inverse A–1.
Formula for the inverse of a 2 × 2 matrix.
A–1
a = 2, b = –1, c = 2, and d = 3
Simplify.
Example 1

13. Multiply A–1 by B to solve the system.
Solve a 2 × 2 System Using an Inverse Matrix
Multiply A–1 by B to solve the system.
X = A–1B
Therefore, the solution of the system is (2, 3).
Answer: (2, 3)
Example 1

14. Write the system in matrix form AX = B.
Solve a 2 × 2 System Using an Inverse Matrix
B. Use an inverse matrix to solve the system of equations, if possible. 2x + y = 9 x – 3y + 2z = 12 5y – 3z = –11
Write the system in matrix form AX = B.
AX = B
Example 1

15. Use a graphing calculator to find A–1.
Solve a 2 × 2 System Using an Inverse Matrix
Use a graphing calculator to find A–1.
A–1
Multiply A–1 by B to solve the system.
Example 1

16. Solve a 2 × 2 System Using an Inverse Matrix
X = A–1B
Answer: (5, –1, 2)
Example 1

17. Use an inverse matrix to solve the system of equations, if possible
Use an inverse matrix to solve the system of equations, if possible. 2x – 3y = –7 –x – y = 1
A. (–2, 1)
B. (2, –1)
C. (–2, –1)
D. no solution
Example 1

18. His collection of coins can be represented by
Solve a 3 × 3 System Using an Inverse Matrix
COINS Marquis has 22 coins that are all nickels, dimes, and quarters. The value of the coins is $2.75. He has three fewer dimes than twice the number of quarters. How many of each type of coin does Marquis have?
His collection of coins can be represented by
n + d + q = 22 5n + 10d + 25q = 275 d – 2q = –3,
where n, d, and q represent the number of nickels, dimes, and quarters, respectively.
Write the system in matrix form AX = B.
Example 2

19. Use a graphing calculator to find A–1.
Solve a 3 × 3 System Using an Inverse Matrix
Use a graphing calculator to find A–1.
A–1
Example 2

20. Multiply A–1 by B to solve the system. A–1B
Solve a 3 × 3 System Using an Inverse Matrix
Multiply A–1 by B to solve the system.
A–1B
Answer: 7 nickels, 9 dimes, and 6 quarters
Example 2

21. MUSIC Manny has downloaded three types of music: country, jazz, and rap. He downloaded a total of 24 songs. Each country song costs $0.75 to download, each jazz song costs $1 to download, and each rap song costs $1.10 to download. In all he has spent $23.95 on his downloads. If Manny has downloaded two more jazz songs than country songs, how many of each kind of music has he downloaded?
A. 6 country, 8 jazz, 10 rap
B. 4 country, 6 jazz, 14 rap
C. 5 country, 7 jazz, 12 rap
D. 7 country, 9 jazz, 9 rap
Example 2

22. Key Concept 3

23. The coefficient matrix is . Calculate the determinant of A.
Use Cramer’s Rule to Solve a 2 × 2 System
Use Cramer’s Rule to find the solution to the system of linear equations, if a unique solution exists. 4x1 – 5x2 = –49 –3x1 + 2x2 = 28
The coefficient matrix is Calculate the determinant of A.
4(2) – (–5)(–3) or –7
Example 3

24. Use Cramer’s Rule to Solve a 2 × 2 System
Because the determinant of A does not equal zero, you can apply Cramer’s Rule.
So, the solution is x1 = –6 and x2 = 5 or (–6, 5). Check your answer in the original system.
Answer: (–6, 5)
Example 3

25. Use Cramer’s Rule to find the solution of the system of linear equations, if a unique solution exists. –6x + 2y = 28 x – 5y = –14
A. no solution
B. (–4, –2)
C. (4, –2)
D. (–4, 2)
Example 3

26. The coefficient matrix is . Calculate the determinant of A.
Use Cramer’s Rule to Solve a 3 × 3 System
Use Cramer’s Rule to find the solution of the system of linear equations, if a unique solution exists. y + 4z = –1 2x – 2y + z = –18 x – 4z = 7
The coefficient matrix is Calculate the determinant of A.
Example 4

27. Formula for the determinant of a 3 × 3 matrix
Use Cramer’s Rule to Solve a 3 × 3 System
Formula for the determinant of a 3 × 3 matrix
Simplify.
Simplify.
Example 4

28. Use Cramer’s Rule to Solve a 3 × 3 System
Because the determinant of A does not equal zero, you can apply Cramer’s Rule.
Example 4

29. Use Cramer’s Rule to Solve a 3 × 3 System
Example 4

30. Therefore, the solution is x = –1, y = 7, and z = –2 or (–1, 7, –2)
Use Cramer’s Rule to Solve a 3 × 3 System
Therefore, the solution is x = –1, y = 7, and z = –2 or (–1, 7, –2)
Answer: (–1, 7, –2)
Example 4

31. Use Cramer’s Rule to Solve a 3 × 3 System
CHECK Check the solution by substituting back into the original system.
7 + 4(–2) = –1 ?
–1 = –1  ?
2(–1) – 2(7) + –2 = –18
–18 = –18  ?
–1 – 4(–2) = 7
7 = 7
Example 4

32. Use Cramer’s Rule to find the solution of the system of linear equations, if a unique solution exists. x – y + 2z = –3 –2x – z = 3 3y + z = 10
A. (2, –3, –1)
B. (–2, 3, 1)
C. (2, 3, 1)
D. no solution
Example 4

33. Solving Linear Systems Using Inverses and Cramer’s Rule
LESSON 6–3
Solving Linear Systems Using Inverses and Cramer’s Rule