If your solution to stated problem does not match the given solution, you should self-critique per instructions at
http://vhcc2.vhcc.edu/dsmith/geninfo/labrynth_created_fall_05/levl1_22/levl2_81/file3_259.htm
.
Your solution, attempt at solution. If you are unable to attempt a solution, give a phrase-by-phrase interpretation of the problem along with a statement of what you do or do not understand about it. This response should be given, based on the work you did in completing the assignment, before you look at the given solution.
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
Your solution:
Confidence Assessment:
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. **
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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. **
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**** 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
Your solution:
Confidence Assessment:
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.
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