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Solve Linear System using Inverse Matrix | Find Cost of Items in Gift Baskets

Learn how to solve a linear system using inverse matrix and find the cost of individual items in different gift baskets.

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Solve Linear System using Inverse Matrix | Find Cost of Items in Gift Baskets

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

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

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

  4. 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

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

  6. Write a system of equations. Let mbe the cost of a movie pass, pbe the cost of a package of popcorn, and dbe the cost of a DVD. STEP 2 Rewrite the system as a matrix equation. STEP 3 2 1 0 2 2 1 4 3 2 15.50 37.00 72.50 m p d = EXAMPLE 5 Solve a multi-step problem 2m + p = 15.50 Equation 1 2m + 2p + d = 37.00 Equation 2 4m + 3p + 2d = 72.50 Equation 3

  7. Enter the coefficient matrix Aand the matrix of constants Binto a graphing calculator. Then find the solution X = A–1B. STEP 4 EXAMPLE 5 Solve a multi-step problem A movie pass costs $7, a package of popcorn costs $1.50, and a DVD costs $20.

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

  9. 11. ANSWER 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? for Examples 4 and 5 GUIDED PRACTICE movie pass: $8 package of popcorn: $1 DVD: $17

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