Alan

"5 pencils and 4 crayons are 26.3 inches. 6 pencils and 5 crayons are 32.0 inches. Each pencil is the same length as the other pencils, and each crayon is the same length as the other crayons. What is the length of 1 pencil? What is the length of 1 crayon?"

5p + 4c = 26.3           ...(1)

6p + 5c = 32              ...(2)

Multiply (1) by 5 and (2) by 4

25p + 20c = 5*26.3     ...(3)

24p + 20c = 4*32        ...(4)

subtract (4) from (3)

p = 5*26.3 - 4*32 =  ...

Substitute the result into (1) or (2) and rearrange to find c.

I'll leave the number crunching to you. 

Here :

P(x) = ax² + bx + c

-3 = a*(0)² + b*(0) + c  so c = -3 immediately

P(x) = ax² + bx -3

13 = a*(-2)² + b*(-2) - 3  or  4a - 2b  = 16     Divide all terms by 2    2a - b = 8    (1)

3 = a*(3)² + b*(3) - 3      or  9a + 3b  = 6      Divide all terms by 3     3a + b = 2   (2)

Now subtract (1) from (2) to find a, then substitute the result back into either (1) or (2) to get b.

Looks good!

So, the solution x/y = -1 is not compatible with part 2.

What about the solution x/y = 3/2?
 

The "multiply by 2" was specific to that question,. The idea was to make the first equation have the same number of b's as the second, so that when (1) was subtracted from (2) the b's would cancel out and there would only be a's and constants left.

Part 1 is as follows:

Let's see you attempt part 2.

As follows:

tan^-1 3 = 71.56º

tan^-1 2 = 63.43º

Work out the cost per paperclip for each box by dividing the number in the box by the cost of the box.  The box with the smaller ratio is the one with the better value for money.

Use the procedure of an almost identical question here: https://web2.0calc.com/questions/help-with-polynomial_1#r1