75c + 100h = 10,000
150c + 75h = 16,875


Here is how I did it and cannot get the second half to check.
75c + 100h = 10,000
150h + 75h = 16,875
225c + 175h = 26,875

225c + 175h = 26,875
225c + 175h - 175h = 26,875 - 175h
225c = 26700
225 225

c= 118.6666667

next

75(118.67) + 100h = 10,000
8900.25 + 100h = 10,000
8900.25-8900.25 + 100h = 10,000 - 8900.25

100 1099.75
100 100

h = 10.9975

-check one-
75(118.67) + 100 (10.9975) = 10,000
8900.25 + 1099.75 = 10,000
10,000 = 10,000

-check two-
150(118.67) + 75(1099.75) = 16,875
1335037.5 + 82481.25 = 1417518.75?? This is wherer I am getting stuck

No, that's not how you do it.

Adding both equations like that doesn't eliminate one of the variables, plus you did a mistake on this line:

225c + 175h - 175h = 26,875 - 175h

This does NOT become

225c = 26700

But this becomes: 225c = 26875 - 175h

Okay, let's start over:

75c + 100h = 10,000
150c + 75h = 16,875

You can actually simplify both equations, but I don't want to confuse you either, so let's take it from there. Multiply the first equation by 2.

2(75c + 100h) = 2(10,000)

150c + 200h = 20,000

See what I did? Now you have 150c in both equations, and you can subtract one equation from the other. Can you take it from here?

150c + 200h = 20,000
150c + 75h = 16,875