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Sec. 7.3c Let A be the coefficient matrix of a system of n linear equations in n variables given by AX = B, where X is the n x 1 matrix of variables and B is the n x 1 matrix of numbers of the right-hand side of the equations. If A–1 exists, then the system of equations has the unique solution X=A –1 B Write the system of equations as a matrix equation AX = B, with A as the coefficient matrix of the system. x 3y z 9 2x 4z 1 8 x y z 5 1 3 1 x 9 AX = B: 2 0 4 y 1 8 1 1 z 5 Write the matrix equation as a system of equations 1 2 3 1 x 2 x 2 y 3 z w 2 0 0 2 8 y 3 2z 8w 3 9 0 1 5 z 9 9x z 5w 9 1 1 6 3 w 2 x y 6 z 3w 2 Solve the given system using inverse matrices 3x 2 y 0 0 x 3 2 X B x y 5 A 5 y 1 1 To solve for X, apply the inverse of A to both sides of the matrix equation: 10 X A B 15 1 3 x 2 y A X B x y Solution: (x, y) = (10, 15) Solve the given system using inverse matrices 3x 3 y 6 z 20 x 3 y 10 z 40 x 3 y 5 z 30 Find 1 XA B Solution: (x, y, z) = (18, 118/3, 14) 3 3 6 x 20 A 1 3 10 X y B 40 30 1 3 5 z Solve the given system using inverse matrices x 4 y 2z 0 2x y z 6 3x 3 y 5 z 13 Find 1 XA B Solution: (x, y, z) = (3, –1/2, 1/2) 1 4 2 x 0 A 2 1 1 X y B 6 3 3 5 z 13 Solve the given system using inverse matrices 2x y 2z 8 3x 2 y z w 10 2 x 3w y 1 4 x 3 y 2 z 5w 39 Find 1 XA B Solution: (x, y, z, w) = (4, –2, 1, –3) 2 1 2 0 x 8 3 2 1 1 y 10 X B A 2 1 0 3 z 1 4 3 2 5 w 39 Use a method of your choice to solve the given system. x yz 6 x y 2 z 2 Augmented Matrix: 1 1 1 6 1 1 2 2 1 0 1.5 2 RREF: 0 1 0.5 4 Solution: (x, y, z) = (2 – 1.5z, –4 – 0.5z, z) Fitting a parabola to three points. Determine a, b, and c so that the points (–1, 5), (2, –1), and (3, 13) are on the graph of f x ax bx c 2 How about a diagram to start??? We need f(–1) = 5, f(2) = –1, and f(3) = 13: f 1 a b c 5 f 2 4a 2b c 1 f 3 9a 3b c 13 Now, simply solve this system!!! (a, b, c) = (4, –6, –5) f x 4x 6x 5 2 Double-check with a graph? Mixing Solutions. Aileen’s Drugstore needs to prepare a 60-L mixture that is 40% acid using three concentrations of acid. The first concentration is 15% acid, the second is 35% acid, and the third is 55% acid. Because of the amounts of acid solution on hand, they need to use twice as much of the 35% solution as the 55% solution. How much of each solution should they use? x = liters of 15% solution y = liters of 35% solution z = liters of 55% solution x y z 60 y 2z 0 0.15x 0.35 y 0.55z 0.40 60 Solve the system!!! Need 3.75 L of 15% acid, 37.5 L of 35% acid, and 18.75 L of 55% acid to make 60 L of 40% acid solution. Manufacturing. Stewart’s metals has three silver alloys on hand. One is 22% silver, another is 30% silver, and the third is 42% silver. How many grams of each alloy is required to produce 80 grams of a new alloy that is 34% silver if the amount of 30% alloy used is twice the amount of 22% alloy used? x = amount of 22% alloy y = amount of 30% alloy z = amount of 42% alloy x y z 80 0.22 x 0.30 y 0.42 z 27.2 2x y 0 Solve the system!!! Need approximately 14.545g of the 22% alloy, 29.091g of the 30% alloy, and 36.364g of the 42% alloy to make 80g of the 34% alloy. Vacation Money. Heather has saved $177 to take with her on the family vacation. She has 51 bills consisting of $1, $5, and $10 bills. If the number of $5 bills is three times the number of $10 bills, find how many of each bill she has. x = number of $1 bills y = number of $5 bills z = number of $10 bills x y z 51 x 5 y 10 z 177 y 3z 0 Solve the system!!! Heather has 27 one-dollar bills, 18 fives, and 6 tens.