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Pre-Calculus Chapter 7 Review Name _________________________________ Date ___________________ Hour ______ 7.1 SUBSTITUTION Solve the system by the method of substitution. Show all work. π₯+π¦ =2 π₯2 β π¦2 = 9 1. { { 2. π₯βπ¦ =0 π₯βπ¦ =1 3. { π¦ = 2π₯ 2 π¦ = π₯ 4 β 2π₯ 2 4. Geometry The perimeter of a rectangle is 480 meters and its length is 1.5 times its width. Find the dimensions of the rectangle. 7.2 ELIMINATION Solve the systems using elimination. Show all work. 1 3 7 2π₯ β π¦ = 2 3π₯ β 2π¦ = 0 π₯+ π¦= 5. { 7. { 5 10 50 6π₯ + 8π¦ = 39 3π₯ + 2(π¦ + 5) = 10 6. { 2 1 1 π₯+ π¦= 5 2 8. { 1.25π₯ β 2π¦ = 3.5 5π₯ β 8π¦ = 14 5 9. Airplane Speed Two planes leave Pittsburgh and Philadelphia at the same time, each going to the other city. One plane flies 25 miles per hour faster than the other. Find the airspeed of each plane if the cities are 275 miles apart and the planes pass each other after 40 minutes of flying time. 7.3 & 7.4 MULTI-VARIABLE SYSTEMS Use back substitution to solve the system of equations. 3π₯ β π¦ + 5π§ = β10 10. { 4π¦ β 2π§ = 16 β3π§ = 18 Determine the order of the following matrices. β9 0 2 6 7 8 6 ] 11. [ 12. [ 5 2 1 0 4 7 0 β8 ] 0 Write the augmented matrix for the system. 5π₯ + 2π¦ + 3π§ = 2 13. { 5π₯ + π¦ =4 8π₯ + 10π§ = β1 Solve each system using Row Echelon Form or Matrices. Include all notations/necessary steps. 4π₯ + 2π¦ + 3π§ = 11 π₯ β 3π¦ + 2π§ = 5 15. { π₯ β 2π¦ + π§ = 6 14. { π₯ + 2π¦ + π§ = 7 2π₯ + π¦ + 2π§ = 7 2π₯ β π¦ + 5π§ = 18 7.5 MATRICES Find x and y. π₯+3 4 0 β3 16. [ β2 π¦+5 β4π¦ 5π₯ β 1 2 ]=[ 0 6π₯ β2 4 β3 16 β44 2 ] 6 Find, if possible, (a) A + B, (b) A β B, (c) 4A, and (d) A + 3B 7 3 10 β20 6 ], π΅ = [ ] 17. π΄ = [ 18. π΄ = [5 β1 5 14 β3 3 0 β1 2 7 0 2], π΅ = [β4 3 2 5 8 β1 1 6] 1 19. Sales At a dairy mart, the numbers of gallons of skim, 2%, and whole milk sold on Friday, Saturday, and Sunday of a particular week are given b the following matrix. Skim 2% Whole milk milk milk 40 π΄ = [60 76 64 82 96 52 πΉπππππ¦ 76] πππ‘π’ππππ¦ 84 ππ’ππππ¦ A second matrix gives the selling price per gallon and the profit per gallon for each of the three types of milk sold by the dairy mart. Selling price per Profit gallon per gallon 2.65 π΅ = [2.81 2.93 0.25 ππππ ππππ 0.30] 2% ππππ 0.35 πβπππ ππππ (a) Find AB. What is the meaning of AB in the context of the situation? (b) Find the dairy martβs profit for Friday through Sunday. 7.6 INVERSE MATRIX Use an inverse matrix to solve (if possible) the system of linear equations. π₯ + 5π¦ = β1 18. { 3π₯ β 5π¦ = 5 3π₯ + 2π¦ β π§ = 6 π₯ 19. { β π¦ + 2π§ = β1 5π₯ + π¦ + π§ = 7 π₯ + 2π¦ + π§ β π€ = β2 2π₯ + π¦ + π§ + π€ = 1 20. { π₯ β π¦ β 3π§ =0 π§+π€ =1 21. Fast-Food Sales A small fast-food chain with restaurants in Santa Monica, Long Beach, and Anaheim sells only hamburgers, hot dogs, and milk shakes. On a certain day, sales ere distributed according to the following matrix.