LESSON 6 3 Solving Linear Systems Using Inverses

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

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

Over Lesson 6 -2 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

Over Lesson 6 -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 + 2 x 2 + 3 x 3 = – 5 2 x 1 + x 2 + x 3 = 1 x 1 + x 2 – x 3 = 8

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

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

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.

• square system • Cramer’s Rule

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

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 = 2, b = – 1, c = 2, and d=3 Simplify.

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). Answer: (2, 3)

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

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.

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

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

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 5 n + 10 d + 25 q = 275 d – 2 q = – 3, where n, d, and q represent the number of nickels, dimes, and quarters, respectively. Write the system in matrix form AX = B.

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

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

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

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

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 x 1 = – 6 and x 2 = 5 or (– 6, 5). Check your answer in the original system. Answer: (– 6, 5)

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

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 + 4 z = – 1 2 x – 2 y + z = – 18 x – 4 z = 7 The coefficient matrix is the determinant of A. . Calculate

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

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.

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

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)

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 ? 2(– 1) – 2(7) + – 2 = – 18 ? – 1 – 4(– 2) = 7 7 = 7

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

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