The difference between the square of two numbers is 11. Twice the square of the first number increased by the square of the second number is 97 find the numbers

Below in bold.

Step-by-step explanation:

x^2 - y^2 = 11

2x^2  + y^2 = 97

From the first equation:

y^2 = x^2 - 11

Substituting in the second equation:

2x^2 + x^2 - 11 = 97

3x^2 = 108

x^2 = 36

x = 6, -6.

Substituting  for x in the first equation:

(6)^2 - y^2 = 11

y^2 = 36 - 11 = 25

y = 5, -5.