short, middle, long pieces of ribbon, totaling 81 inches.
long = short + 34
long = 2 * middle
short + middle + long = 81
this is called a "system of equations" with 3 variables (short, middle, long)
to solve a system of equations with 3 variables, you need 3 equations. We have 3 equations, so we should be good.
in math terms, we're going to use the "substitution method"
so here's our equations:
long = short + 34
long = 2 * middle
short + middle + long = 81
since long = short + 34, we'll replace long with short + 34 in the other two equations like this:
short + 34 = 2 * middle
short + middle + short + 34 = 81
let's simplify
short + 34 = 2 * middle
2 * short + middle + 34 = 81
let's get the 34s to the other side of both equations by subtracting 34 from both sides like this:
short = 2 * middle - 34
2 * short + middle = 81 - 34
simplify
short = 2 * middle - 34
2 * short + middle = 47
in the bottom equation let's get middle to the other side:
short = 2 * middle - 34
2 * short = 47 - middle
in the bottom equation let's get short by itself.
short = 2 * middle - 34
short = (47 - middle) / 2
since short = short, we can say this:
2 * middle - 34 = (47 - middle) / 2
4 * middle - 68 = 47 - middle
5 * middle - 68 = 47
5 * middle = 115
middle = 23
but short = 2 * middle - 34 so:
short = 2 * 23 - 34
short = 46 - 34
short = 12
but short + middle + long = 81 so:
12 + 23 + long = 81
long + 35 = 81
long = 46
Answer: short = 12, middle = 23, long = 46