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

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

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.