#$&* course mth 164 SOLUTIONS/COMMENTARY ON QUERY 14 **** Query problem 10.1.22 (5th ed 10.1.24) solve the equation 3x - y = 7, 9x - 3y = 21.
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14:24:23 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: We can look at the two equations and see that the second one is a multiple of the first so therefore they will be equal to each other and also meaning that the graphs will coincide. confidence rating #$&*:3 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: ** The two equations coincide. If we subtract 3 times the first from the second we get 0 = 0. This tells us that one equation is a multiple of the other, and that they are therefore equivalent. Their graphs coincide. A solution to one equation is a solution to the other. So the solution set consists of all (x, y) satisfying the first equation 3 x - y = 7. These solutions lie on the line y = 3x - 7. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*:OK Query problem 10.2.12 (5th edition 10.1.42) solve the equation 3x-2y+2z=6, 7x-3y+2x=-1, 2x-3y+4z=0. ** That's 2z in the second equation, not 2x. ** ** If we row reduce the coefficient matrix [[3,-2,2,6],[7,-3,2,-1],[2,-3,4,0]] we get [[1, 0, -0.4] [0; 0, 1, -1.6] [0; 0, 0, 0, 1]], indicating inconsistent system. Using the equations: From the equations 3 x - 2 y + 2 z = 6 and 2 x - 3 y + 4 z = 0 we can eliminate z (add -2 times the first to the second) to get -4 x + y = -12 or y = 4x - 12. From the equations 3 x - 2 y + 2 z = 6 and 7x-3y+2z=-1 we eliminate z (just subtract the equations) to get y = 4 x + 7 . y cannot be equal to 4x - 12 at the same time it's equal to 4x + 7. The two expressions could only be equal if -12 = 7. We conclude that the system is inconsistent. No solution exists. The first thing I did was number the equations as follows: x - y + z = -4 (Equ #1) 2x -3y + 4z = -15 (Equ #2) 5x +y - 2z = 12 (Equ #3) then, I replaced #1 with product of #1 & -2 making it -2x + 2y -2z = 8, then I replaced #2 with the sum of #1 & #2 making #2 : -y + 2z = -7, then I multiplied #1 by 5 and multiplied #3 by 2 making #1 -10x + 10y - 10z = 40 and #3: 10x + 2y - 4z = 24, next I replaced #3 with the sum of #1 & #3 making #3: 12y - 14z = 64, then I multiplied #2 by 6 and multiplied #3 by 1/2 making #2: -6y = 12z = -42 and #3: 6y -7z = 32, I then replaced #3 with the sum of #2 & #3 making #3 5z = -10 which then solves to z = -2, Next I back substituted -2 for z in #1 & #2 making #1: -10x + 10y = 20 and #2: y = 3, I then substituted 3 for y in #1 and solved it for x: x = 1. ** Your system differs from the one in my notes. Your solution x = 1, y = 3, z = -2 is correct for your equations, which I suspect are the correct equations from the book. ** ** Solution to the system as given in my note as x - y - z = -4, 2•x - 3•y + 4•z = -15, 5•x + y - 2•z = 12: We start with x - y - z = -4 2x -3y + 4z = -15 5x +y - 2z = 12. We multiply the first equ by -1 and add it to the second: -2x + 2y + 2z = 8 2x - 3y + 4z = -15. _______________________ -y + 6z = -7 Add -5 times first to third to get -5x + 5y + 5z = 20 5x + y - 2z = 12 _______________________ 6y + 3 z = 32. Solve the system -y + 2z = -7, 6y - 7z = 32. First add 6 times first to second: -6y + 36 z = -42 6y + 3 z = 32 ________________________ 39 z = -10 z = -10/39. Substituting into the equation 6y + 3z = 32 we get 6y + 3 * -10/39 = 32 which gives solution y = 71/31. Substituting z = -10/39 and y = 71/13 into the first original equation x - y - z = -4 we get x = 47/39. ** &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): This one is completely mixed up I cannot tell what is what and I was reading the given solution before I knew it. ------------------------------------------------ Self-critique rating #$&*: **** Query problem 10.1.44 (5th ed 10.1.60) ticket $9 adult $7 senior, 325 people paid $2495.
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14:38:27 YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: 9x + 7y = 2495 X = y = 325 -9x – 9y = -2925 -2y = -430 Y = 215 215 senior citizens 110 adults confidence rating #$&*:2 ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: Good student solution: We can set up a formula for the price of the tickets by saying x= adult ticket price and y=senior citizen ticket price so we have 9x+7y=2495 We can then write an equation for the amount of tickets purchased by saying x+y=325 We can then multiply the second equation by -9 and we get -9x-9y=-2925 so when we add the sum of these two equations we get -2y=-430 so y=215 so there were 215 senior citizens and 110 adults. INSTRUCTOR COMMENT: Good. x adults and y seniors at $9 per adult and $7 per senior yields 9 x + 7 y dollars. This is equal to the $2495, giving us the equation 9x + 7y = $2495. The total number of tickets is x + y = 325, giving us the second equation. This equation is solved as indicated by the student. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating #$&*: