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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

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

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

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

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

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

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

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

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

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

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

square system Cramer’s Rule Vocabulary

Key Concept 1

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

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

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: Example 1

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

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

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

X = A–1B Answer: Solve a 2 × 2 System Using an Inverse Matrix Example 1

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

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

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

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

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

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: Example 2

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

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

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

Key Concept 3

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

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: Example 3

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

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

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

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

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

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

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

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: Example 4

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

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

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

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

End of the Lesson