Question

find the 5 diff. ways a collection of 100 coins-pennies, dimes, and quarters- can be worth exactly worth $4.99.

Answer 1

p + d + q = 100
p + 10d + 25q = 499

subtract them to get 9d + 24q = 399 or
3d + 8q = 133

we need values of d and q which are positive whole numbers.

After a few trials I got
d = 7 and q = 14

the "slope" of 3d + 8q = 133 is - 3/8
that is, if we increase the value of d by 8 and decrease the value of q by 3 we get another whole number solution.

so I have the following ordered pairs of (d,q)
7 14
15 11
23 8
31 5
39 2

sub each of these into my first equation to find p (pennies)

I will do one of them, you can do the rest
if d=23 q=8 then
p+23+8 = 100
p = 69
check: 23+8+69 = 100
69 + 10(23) + 25(8) = 499