Presentation on theme: "Use an inverse matrix to solve the linear system."— Presentation transcript:
1 Use an inverse matrix to solve the linear system. EXAMPLE 4Solve a linear systemUse an inverse matrix to solve the linear system.2x – 3y = 19Equation 1x + 4y = –7Equation 2SOLUTIONSTEP 1Write the linear system as a matrix equationAX = B.coefficient matrix of matrix of matrix (A) variables (X) constants(B)2 –31 4.xy19–7=
2 EXAMPLE 4Solve a linear systemSTEP 2Find the inverse of matrix A.4 3–1 2=A–1 =18 – (–3)41132–STEP 3Multiply the matrix of constants by A–1 onthe left.X = A–1B =41113–219–7=5–3xy
3 EXAMPLE 4Solve a linear systemThe solution of the system is (5, – 3).ANSWERCHECK2(5) – 3(–3) = = 195 + 4(–3) = 5 – 12 = –7
4 EXAMPLE 5Solve a multi-step problemGiftsA company sells three types of movie gift baskets. A basic basket with 2 movie passes and 1 package of microwave popcorn costs $ 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 $ Find the cost of each item in the gift baskets.
5 EXAMPLE 5Solve a multi-step problemSOLUTIONSTEP 1Write verbal models for the situation.
6 Solve a multi-step problem EXAMPLE 5Solve a multi-step problemSTEP 2Write 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.2m + p = 15.50Equation 12m + 2p + d = 37.00Equation 24m + 3p + 2d = 72.50Equation 3STEP 3Rewrite the system as a matrix equation.mpd15.5037.0072.50=
7 EXAMPLE 5Solve a multi-step problemSTEP 4Enter the coefficient matrix A and the matrix of constants B into a graphing calculator. Then find the solution X = A–1B.A movie pass costs $7, a package of popcorn costs$1.50, and a DVD costs $20.
8 GUIDED PRACTICEfor Examples 4 and 5Use an inverse matrix to solve the linear system.4x + y = 103x + 5y = –18.9.2x – y = – 66x – 3y = – 1810.3x – y = –5–4x + 2y = 8(3, –2).ANSWERANSWER(– 1, 2)ANSWERinfinitely many solutions
9 GUIDED PRACTICEfor Examples 4 and 511.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?ANSWERmovie pass: $8package of popcorn: $1DVD: $17