1. Lesson Menu 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

2. Over Lesson 6-2 5–Minute Check 1 A.BA is not possible; AB = B.AB is not possible; BA = C.AB = ; BA = D.AB is not possible; BA is not possible Find AB and BA, if possible.

3. Over Lesson 6-2 5–Minute Check 2 Write the system of equations as a matrix equation, AX = B. Then use Gauss-Jordan elimination on the augmented matrix to solve for X. x 1 + 2x 2 + 3x 3 = –5 2x 1 + x 2 + x 3 = 1 x 1 + x 2 – x 3 = 8

4. Over Lesson 6-2 5–Minute Check 2 A. ; (1, 3, –4) B. ; (–1, –3, 4) C. ; (1, 3, –4) D. ; (–1, –3, 4)

5. Over Lesson 6-2 5–Minute Check 3 For and, find AB and BA and determine whether A and B are inverse matrices. A. B. C. D.

6. Over Lesson 6-2 5–Minute Check 4 A.0 B.13 C.15 D.17 Which of the following represents the determinant of ?

7. Then/Now 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.

8. Vocabulary square system Cramer’s Rule

9. Key Concept 1

10. Example 1 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.

11. Example 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. a = 2, b = –1, c = 2, and d = 3 Simplify. Formula for the inverse of a 2 × 2 matrix. A –1

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

13. Example 1 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

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

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

16. Example 1 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

17. Example 2 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.

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

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

20. Example 2 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

21. Key Concept 3

22. Example 3 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. 4x 1 – 5x 2 = –49 –3x 1 + 2x 2 = 28 The coefficient matrix is. Calculate the determinant of A. 4(2) – (–5)(–3) or –7

23. Example 3 Use Cramer’s Rule to Solve a 2 × 2 System Answer: (–6, 5) Because the determinant of A does not equal zero, you can apply Cramer’s Rule. So, the solution is x 1 = –6 and x 2 = 5 or (–6, 5). Check your answer in the original system.

24. Example 3 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)

25. Example 4 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.

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

27. Example 4 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.

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

29. Example 4 Answer:(–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)

30. Example 4 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

31. Example 4 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