Optimization Question: values for x, y, z which give highest value for "a"

bluejelly

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Jan 15, 2016
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Hi

I pretty sure this is a simple one but I've forgotten most of the useful math I learnt in School / University :(

I have four equations and I'm interested to find out what values of x y and z will produce the highest value for a

3x - y - z = a

4y - x - z = a

3z - x - y = a

x + y + z =100

Just to clarify, x, y, z and a will have the same value in each equation.

If anyone could give me pointers it would be appreciated

Thanks!
 
I have four equations and I'm interested to find out what values of x y and z will produce the highest value for a

3x - y - z = a

4y - x - z = a

3z - x - y = a

x + y + z =100
This is a system of four linear equations in four variables. One can solve it for the specific values of the variables (if the system if solvable), but this is not an optimization exercise (because these are not inequalities). And I'm not seeing how this is calculus, since derivatives, etc, would not seem to have any application here.

Instead, just use what you learned back in algebra to solve the system of equations. Or you can use "technology", like a graphing calculator, to solve the system. But remember to enter the system in a format that the technology can understand, such as:

. . . . .+3x1y1z1a=0\displaystyle +3x\, -\, 1y\, -\, 1z\, -\, 1a\, =\, 0

. . . . .1x+4y1z1a=0\displaystyle -1x\, +\, 4y\, -\, 1z\, -\, 1a\, =\, 0

. . . . .1x1y+3z1a=0\displaystyle -1x\, -\, 1y\, +\, 3z\, -\, 1a\, =\, 0

. . . . .+1x+1y+1z+0a=100\displaystyle +1x\, +\, 1y\, +\, 1z\, +\, 0a\, =\, 100

Then the matrix to reduce would be:

. . . . .[3111014110113101110100]\displaystyle \left[\, \begin{array}{rrrrr}3&-1&-1&-1&0\\-1&4&-1&-1&0\\-1&-1&3&-1&0\\1&1&1&0&100\end{array}\, \right]

This system is solvable, by the way, with unique (not "optimized" or "highest") values for each of the variables. ;)
 
Hi

I pretty sure this is a simple one but I've forgotten most of the useful math I learnt in School / University :(

I have four equations and I'm interested to find out what values of x y and z will produce the highest value for a

3x - y - z = a

4y - x - z = a

3z - x - y = a

x + y + z =100

Just to clarify, x, y, z and a will have the same value in each equation.

If anyone could give me pointers it would be appreciated

Thanks!
Saying the same thing in a different way than what staple said: If x, y, z, and a will have the same value for each equation then the system has a solution for a unique a which then would determine x, y, and z. That a is 3007\displaystyle \frac{300}{7}.
 
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