The sum of two numbers is 33 and their difference is 7. Find the sum of their…

Solution:

Let the two numbers be x and y. We are given x + y = 33 and x - y = 7.

1. Add the equations: (x + y) + (x - y) = 33 + 7, so 2x = 40 and x = 20.

2. Subtract the equations: (x + y) - (x - y) = 33 - 7, so 2y = 26 and y = 13.

Now compute the sum of their squares: x^2 + y^2 = 20^2 + 13^2 = 400 + 169 = 569.

Alternative method:

• Use the identity (x + y)^2 + (x - y)^2 = 2(x^2 + y^2). Substitute the given sums: 33^2 + 7^2 = 1089 + 49 = 1138.

• Divide by 2 to get x^2 + y^2 = 1138 / 2 = 569.

Final answer: 569