Question

Three different groups of friends decide to meet up and catch a popular motion picture at a local movie theater. The first group buys 4 tickets, 3 drinks, and 5 orders of popcorn for a total of $85. A second group buys 3 tickets, 3 drinks, and 2 orders of popcorn for a total of $57. The third group buys 4 tickets, 5 drinks, and 3 orders of popcorn for a total of $83. How much does a ticket, drink, and popcorn cost? For the purpose of simplicity, assume that there is only one type of ticket (e.g., no difference in child, adult, elderly ticket costs), and drinks and popcorn come in one size.

Part 1. [1 Point] After reading the problem description, model a system of three linear equations with three variables and represent it as an augmented matrix.

Part 2. [1 Point] Determine the three unknown variables from the model generated in part 1 by using the row-echelon form approach. Shown your work.

Part 3. [1 Point] Find the determinant of the A matrix in part 1 using cofactor expansion to prove that the A matrix is invertible. Show your work and explain your final answer. Note: The A matrix does not include the solution vector.

Part 4. [1 Point] Find the inverse of the A matrix in part 1 by using the row-echelon form approach. Show your work and explain your final answer.

Part 5. [1 Point] Different estimation approaches that yield unknown variables could be exercised for verification purposes. Use the inverse matrix found in part 4 to solve the system of linear equations that were modeled in part 1. The results should be consistent with the results found in part 2

Answer 1

tick. ticket het a drink 4 and 2 be the cost of and popcorn, respectively, & 4x+3y +52 85 PartL 3x+3y + 2 z 57 First Group Second Group Third Group - 4x + 5y + 32 = 83 The of the above system augmented matrix [A 67 Part? >> Мол 4 3 5 85 3 3 2 57 4 5 3 83 R, R, - R2 R3 R3-R2 103 28 57 26 12 R2 ~ R₂ - 3 Ri Rz → R₃-RI 0 برای SHOT ON REDMI Y3 0 -2 28 -27 -2 Riz → R3 / 2 o 3 1 28 -7 .-27 1 -1 -1 0 دی ۔ Rough

R₂ R₂ - 3R3 CLASSTIME Dare 1-24 - -1 R24 0000 Re (-4) T O 3 1 28 ] . R2 PR3. R2 R R₂ + Rz. R, - 3R3 Lol.15 5 > X = 10 x = lo y 25 z Z z 6 . Az [ 4 3 5 1 3 3 .2 14 5 3 1 det (A) & Expansion along first row. 2 4/9 - 10 -3 (9-8) + 5 (1-12) 2 4 0-1 -3 +5x3 -4-3 + 15 15 - 7 NNNN det (A) 2 det (A) to A is non-singular & Inverse of t enists, os

Date А е т д II ws DO- 3 2 5 3 R, - R -R2 R3 R3-R2 - - - w [RED] =[A] lo 3 -1 0 3 3 2 Lo Io 2 0- R2 ~ R₂ - BR R3 R₃ R - w 3 2 -7 -3 4 0 -2 1- 0 1 R3 R3/2 o 3 -1 0 3 -7 1-3 4 o -1 -/ 2012 Re - R₂ - 3 R3 o 1 0 o 0 SHOT ON REDMI YS! 1 3 -4 -1 -3/ 1-1/ - 4 0 0 -3/2 / O AI DUAL CAMERA

(CLASSTIME Page Date TO 3 R2 Rz 1 0 Lo 0 1 o R, R2 3 - -1 -1/2 0 " 1 3/8 -1 R -3R3. Rat Rz. 0 V2 3/3) 0 J 0 Lolo 1 -19 1-1/8 13/8 2 -1 -1 9/8 7/8 /8 3 2 21-182 - 918 - 1 718 3/8 - 3%8 the 1 system of by inverse Parts now solving equations in part method, А х 2 ь » az Alb -1/82-9/ -18 4 7/8 L3/8 - 3/8 57 183

-557x2 - 4x4 O AI DUAL CAMERA SHOT ON REDMI Y3 (-25-57+SET) I (255 -57+ 249 ) (114 - 832/ 8 7 490 - 57) (504-57) To 114 - 104 62-57 63-57 Therefore, the solution to the of equation is unique and (x, y, z) = (1o, s, 6) given system same by