EXAMPLE 4 Solve a linear system Use an

EXAMPLE 4 Solve a linear system STEP 2 Find the inverse of matrix A.

EXAMPLE 4 Solve a linear system ANSWER The solution of the system is (5,

EXAMPLE 5 Solve a multi-step problem Gifts A company sells three types of movie

EXAMPLE 5 Solve a multi-step problem SOLUTION STEP 1 Write verbal models for the

EXAMPLE 5 STEP 2 STEP 3 Solve a multi-step problem Write a system of

EXAMPLE 5 STEP 4 Solve a multi-step problem Enter the coefficient matrix A and

GUIDED PRACTICE for Examples 4 and 5 Use an inverse matrix to solve the

GUIDED PRACTICE for Examples 4 and 5 11. What if? In Example 5, how

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EXAMPLE 4 Solve a linear system Use an inverse matrix to solve the linear system. 2 x – 3 y = 19 Equation 1 x + 4 y = – 7 Equation 2 SOLUTION STEP 1 Write the linear system as a matrix equation AX = B. coefficient matrix (A) matrix of variables (X) matrix of constants(B) 2 – 3 x 19 1 4 . y = – 7

EXAMPLE 4 Solve a linear system STEP 2 Find the inverse of matrix A. A– 1 1 = 8 – (– 3) 4 3 – 1 2 4 11 = – 1 11 3 11 2 11 STEP 3 Multiply the matrix of constants by A– 1 on the left. X= A– 1 B = 4 11 – 1 11 3 11 2 11 19 – 7 = 5 – 3 = x y

EXAMPLE 4 Solve a linear system ANSWER The solution of the system is (5, – 3). CHECK 2(5) – 3(– 3) = 10 + 9 = 19 5 + 4(– 3) = 5 – 12 = – 7

EXAMPLE 5 Solve a multi-step problem Gifts A company sells three types of movie gift baskets. A basic basket with 2 movie passes and 1 package of microwave popcorn costs $15. 50. A medium basket with 2 movie passes, 2 packages of popcorn, and 1 DVD costs $37. A super basket with 4 movie passes, 3 packages of popcorn, and 2 DVDs costs $72. 50. Find the cost of each item in the gift baskets.

EXAMPLE 5 Solve a multi-step problem SOLUTION STEP 1 Write verbal models for the situation.

EXAMPLE 5 STEP 2 STEP 3 Solve a multi-step problem Write a system of equations. Let m be the cost of a movie pass, p be the cost of a package of popcorn, and d be the cost of a DVD. 2 m + p = 15. 50 Equation 1 2 m + 2 p + d = 37. 00 Equation 2 4 m + 3 p + 2 d = 72. 50 Equation 3 Rewrite the system as a matrix equation. 2 1 0 m 2 2 1 p 4 3 2 d 15. 50 = 37. 00 72. 50

EXAMPLE 5 STEP 4 Solve a multi-step problem Enter the coefficient matrix A and the matrix of constants B into a graphing calculator. Then find the solution X = A– 1 B. A movie pass costs $7, a package of popcorn costs $1. 50, and a DVD costs $20.

GUIDED PRACTICE for Examples 4 and 5 Use an inverse matrix to solve the linear system. 8. 4 x + y = 10 3 x + 5 y = – 1 ANSWER (3, – 2). 9. 2 x – y = – 6 6 x – 3 y = – 18 10. 3 x – y = – 5 – 4 x + 2 y = 8 ANSWER infinitely many solutions ANSWER (– 1, 2)

GUIDED PRACTICE for Examples 4 and 5 11. What if? In Example 5, how does the answer change if a basic basket costs $17, a medium basket costs $35, and a super basket costs $69? ANSWER movie pass: $8 package of popcorn: $1 DVD: $17