Explanation Given: Total number of seats in three parts is 1,000. The revenue collected on all sold seats must be $50,000, where orchestra seats cost $80, main seats cost $50 and balcony seats cost $40. The revenue is $42,500 if all the orchestra and balcony seats are sold and 3 4 of the main seats are sold. Procedure used: 1. Form an augmented matrix [ A | I ] where a matrix A is placed along with a same dimension identity matrix I separated by a line. 2. Apply row operations on [ A | I ] in order to obtain another augmented matrix [ I | B ] where I , is again an identity matrix and B is an inverse of A. 3. If the reduction process yields a row of zeros in the left half of the augmented matrix, then the inverse of A does not exist. Calculation: Let x , y and z be the number of seats in each part that costs $80, $50 and $40, respectively. The total number of seats in three parts is 1,000, the first equation is x + y + z = 1 , 000 . The revenue collected on all sold seats is $50,000, hence the second equation is 80 x + 50 y + 40 z = $ 50 , 000 . The third equation is formulated as, 80 x + 3 4 ( 50 y ) + 40 z = $ 42 , 500 or 80 x + 75 y 2 + 40 z = $ 42 , 500 . Thus, the system of equations becomes, { x + y + z = 1 , 000 80 x + 50 y + 40 z = $ 50 , 000 80 x + 75 2 y + 40 z = $ 42 , 500 Convert the system of equation into a matrix notation. [ 1 1 1 80 50 40 80 75 2 40 ] [ x y z ] = [ 1 , 000 50 , 000 42 , 500 ] Take A = [ 1 1 1 80 50 40 80 75 2 40 ] and X = [ x y z ] and transform [ 1 1 1 80 50 40 80 75 2 40 ] [ x y z ] = [ 1 , 000 50 , 000 42 , 500 ] as follows. A − 1 A X = A − 1 [ 1 , 000 50 , 000 42 , 500 ] ( Multiply with A − 1 on RHS of both sides ) I X = A − 1 [ 1 , 000 50 , 000 42 , 500 ] ( ∵ A − 1 A = I ) X = A − 1 [ 1 , 000 50 , 000 42 , 500 ] Obtain the augmented matrix as follows. [ 1 1 1 1 0 0 80 50 40 0 1 0 80 75 2 40 0 0 1 ] Apply the row operation 2 R 3 → R 3 and reduce the matrix. [ 1 1 1 1 0 0 80 50 40 0 1 0 160 75 80 0 0 2 ] Apply the row operation 80 R 1 − R 2 → R 2 and 160 R 1 − R 3 → R 3 and reduce the matrix

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ISBN: 8220102019737

Author: YOCCO

Publisher: PEARSON

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Chapter 7.4, Problem 32E

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EBK COLLEGE ALGEBRA IN CONTEXT

Ch. 7.1 - In Exercises 14, solve the systems of equations....Ch. 7.1 - Prob. 2E Ch. 7.1 - Prob. 3E Ch. 7.1 - Prob. 4E Ch. 7.1 - Prob. 5E Ch. 7.1 - Prob. 6E Ch. 7.1 - Prob. 7E Ch. 7.1 - Prob. 8E Ch. 7.1 - Prob. 9E Ch. 7.1 - Prob. 10E

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The number of seats in theater.

Solution Summary: The author calculates that the total number of seats in a theater is 1,000. The first equation is x+y+z=1,000, the second equation 80+50y +40z = 50,000, and the third

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