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July 2nd, 2017, 03:34 PM   #1
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Stuck on a Gauss-Jordan Word Problem

I'm stuck on one of the problems. A similar question from the book asks:

An electronics company produces three models of stereo speakers, models A, B and C, and can deliver them by truck, van or SUV. A truck holds 2 boxes of model A, 2 boxes of model B and 3 boxes of model C. A van holds 3 boxes of model A, 4 boxes of model B and 2 boxes of model C. An SUV holds 5 boxes of model A, 7 boxes of model B, and 1 box of model C.

There are 2 parts to this question and I don't understand part b.

b) Model C has been discontinued. If 25 boxes of model A and 33 boxes of model B, how many vehicles of each type should be used to operate at full capacity?

I understand the math behind the problem but, am having trouble wrapping my head around how they arrived at the solution.

Solving the matrix gives:

1 0 0 2

0 1 0 2

0 0 1 3

Since model C is discontinued, you'll have 2 equations. Solving this matrix gives:

1 0 -1/2 1/2

0 1 2 8

So x = 1/2z + 1/2 and y = 8 - 2z

It says "This equation shows that z must be an odd number for x to be a whole number and not a fraction."

Why must z be an odd number?

Solving the inequality:

8 - 2z > 0

z < 4

So z must be 1 or 3. Plug z into the equation when z = 1 to solve for x and y, gives you:

x = 1 and y = 6

and when z = 3:

x = 2 and y = 2

Therefore, the solution using the most trucks is to use 2 trucks, 2​ vans, and 3 SUVs. The other solution is to use 1 trucks, 2​ vans, and 3 SUVs.

I don't understand where these values are coming from.

Last edited by skipjack; July 2nd, 2017 at 04:50 PM.
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July 2nd, 2017, 06:01 PM   #2
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I assume there are x trucks, y vans and z SUVs.
Boxes of model A delivered = 2x + 3y + 5z = 25
Boxes of model B delivered = 2x + 4y + 7z = 33
Those equations can easily be solved to give x = (1/2)z + 1/2 and y = 8 - 2z.

The equation x = 1/2z + 1/2 is equivalent to z = 2x - 1.
Hence if x is an integer, z is odd.

If z is odd, y = 8 - 2z cannot be zero (as that would require z = 4).
Hence if y is non-negative, y > 0, and so 8 - 2z > 0, which implies z < 4.

Hence if z is non-negative (as well as being odd), it is 1 or 3.

If z = 1, x = 1/2 + 1/2 = 1 and y = 8 - 2z = 6.
If z = 3, x = 3/2 + 1/2 = 2 and y = 8 - 2z = 2.

In summary (x, y, z) = (1, 6, 1) or (2, 2, 3).

Although the question didn't ask for the information,
the solution using the most trucks is (x, y, z) = (2, 2, 3).

The other solution, (x, y, z) = (1, 6, 1) uses 1 truck, 6​ vans, and 1 SUV, contrary to what you posted.
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July 2nd, 2017, 07:54 PM   #3
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Thank you for you answer! This problem is much more clear to me. I just have a few additional questions:

1. How do we know z is odd? I plugged in some values into z = 2x - 1 and got odd numbers. Is this how we would know or is there a more obvious way?
2. Also I'm a little confused as to how you knew z couldn't equal 4 before solving the inequality.

Last edited by skipjack; July 3rd, 2017 at 11:26 AM.
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July 3rd, 2017, 11:19 AM   #4
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If x is an integer, 2x is even, and so 2x - 1 is odd.

As z is odd (assuming that x is an integer) and 4 is even, z cannot equal 4.
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July 3rd, 2017, 12:17 PM   #5
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Thank you sooo much!
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