Question

one number is 6 more than another. The difference between their squares is 84. What are the numbers?

Answer 1

X+6=y
Y^2-x^2=84

So substitute.
Y^2-x^2=84
(X+6)^2-x^2=84
X^2+12x+36-x^2=84
12x+36=84
12x=48
X=4

So then use this value to find y by substituting.

X+6=y
4+6=y
10=y

Answer 2

let the number be x and y
x = y + 6
x² - y² = 84       

solve for y, so y = x-6 
Put it in the second eq.
x² - (x-6)² = 84
x² - (x-6)(x-6) = 84
x² -( x²- 12x + 36) = 84
x² - x² + 12x - 36 = 84
12x - 36 = 84
12x = 84 + 36
12x = 120
x = 120/12 = 10
Put this x value in x = y + 6
10 = y + 6
y = 4
Therefore, the numbers are 10 and 4.