Let's use 'x' for the first side, 'y' for the second, and 'z' for the third. Translate the statements into equations:
"The first side of a triangle is twice the second"
x = 2y
"the third is 20 feet less than three times the second"
z = 3y - 20
"the perimeter is 106 feet"
x + y + z = 106
Now let's plug in what 'x' and 'z' equal in the first two equations into the last equation:
2y + y + 3y - 20 = 106
Combine like terms(2y + y + 3y = 6y):
6y - 20 = 106
Add 20 to both sides:
6y = 126
Divide 6 to both sides:
y = 21
So the length of the second side is 21 feet. We can plug this into the first two equations to find the lengths of the other sides:
x = 2y
x = 2(21)
Multiply:
x = 42
So the length of the first side is 42 feet.
z = 3y - 20
z = 3(21) - 20
Multiply:
z = 63 - 20
Subtract:
z = 43
So the length of the third side is 43 feet.