Question

I would be super grateful if someone would help me with these two questions. Attachment below.

Answer 1

You are given that:
\(\color{#000000 }{ \displaystyle -x+6\le y\le x+4 }\)
\(\color{#000000 }{ \displaystyle y\le 7 }\)
\(\color{#000000 }{ \displaystyle x\le 5 }\)

And the expression is:
\(\color{#000000 }{ \displaystyle K=5x-8y }\)
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You are subtracting 8y, and the smaller this 8y, the bigger value you get for K. So, you would plug in the minimum for y, into the equation.

\(\color{#000000 }{ \displaystyle K=5x-8(-x+6) }\)
\(\color{#000000 }{ \displaystyle K=5x+8x-48}\)
\(\color{#000000 }{ \displaystyle K=13x-48}\)

Now, you have a positive component, 13x, and the bigger 13x is, the bigger K you have. You are given that x is at most 5, so to get the maximum for K, what do you have to do?

Answer 2

Now, as far as the second problem goes...

The expression:
\(\color{#000000 }{ \displaystyle L=26-10x+3y }\)

Constraints: (and you want to maximize the L)
\(\color{#000000 }{ \displaystyle x\le 7 }\)

\(\color{#000000 }{ \displaystyle 5-x\le y }\)

\(\color{#000000 }{ \displaystyle y \le x-1 }\)
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The biggest value for y is x-1, so the bigger 3y, the bigger L you get. Therefore, you need to plug in x-1 for y.
\(\color{#000000 }{ \displaystyle L=26-10x+3(x-1) }\)
\(\color{#000000 }{ \displaystyle L=26-10x+3x-3 }\)
\(\color{#000000 }{ \displaystyle L=23-7x }\)

Then, you are subtracting 7x from the expression, and the less you subtract, the bigger value you get. So you would like to have the 7x, as small as posssible, or, for same purpose, you can find the largest value for -7x.

\(\color{#000000 }{ \displaystyle 5-x\le y }\)
\(\color{#000000 }{ \displaystyle -x\le y-5 }\)
\(\color{#000000 }{ \displaystyle -7x\le 7y-35 }\)

And to maximume 7y, (which is part of maximizing -7x) plug in the biggest value the y can have.
\(\color{#000000 }{ \displaystyle y\le x-1 }\)

\(\color{#000000 }{ \displaystyle -7x\le 7(x-1)-35 }\)
\(\color{#000000 }{ \displaystyle -7x\le 7x-36 }\)

Now, you need to plug in (to the right side), the maximum value for x, to find the most/biggest value that -7x can have:
\(\color{#000000 }{ \displaystyle x\le 7 }\)
\(\color{#000000 }{ \displaystyle -7x\le 7(7)-36 }\)
\(\color{#000000 }{ \displaystyle -7x\le 49-36 }\)
\(\color{#000000 }{ \displaystyle -7x\le 13 }\)

Now, you need to finish maximuming the L.
(Recall we had \(\color{#000000 }{ \displaystyle L=23-7x }\))

We have found that -7x is at most 13....

Answer 3

Thank you so much for your help! I am so sorry for not responding my internet isn't working well and I didn't realize anyone had responded to my question. @SolomonZelman