x,y - the numbers
one positive number is 5 less than twice a second number
(1) x - 5 = 2y
their product is 117
(2) xy = 117
(1) x - 5 = 2y add 5 to both sides
x = 2y + 5 substitute it to (2)
(2y + 5)y = 117 use distributive property
(2y)(y) + (5)(y) = 117
2y² + 5y = 117 subtract 117 from both sides
2y² + 5y - 117 = 0
2y² + 18y - 13y - 117 = 0
2y(y + 9) - 13(y + 9) = 0
(y + 9)(2y - 13) = 0 ↔ y + 9 = 0 ∨ 2y - 13 = 0
y + 9 = 0 subtract 9 from both sides
y = -9
2y - 13 = 0 add 13 to both sides
2y = 13 divide both sides by 2
y = 6.5
substitute the values of y to (1)
x = 2(-9) + 5 = -18 + 5 = -13 < 0
x = 2(6.5) + 5 = 13 + 5 = 18
