Word Problem:
There are two numbers. The sum of their squares is 61, and the difference of their squares is 7. What are the two numbers?
Answer Key:
Let's assume that the two numbers are x and y. We can write two equations based on the given information:
x^2 + y^2 = 61
x^2 - y^2 = 7
We can use the second equation to solve for one variable in terms of the other:
x^2 = y^2 + 7
Then we can substitute this expression into the first equation:
(y^2 + 7) + y^2 = 61
Simplifying this equation, we get:
2y^2 + 7 = 61
2y^2 = 54
y^2 = 27
y = sqrt(27)
y = 3*sqrt(3)
Now we can substitute this value of y into one of the previous equations to solve for x:
x^2 + (3*sqrt(3))^2 = 61
x^2 + 27 = 61
x^2 = 34
x = sqrt(34)
Therefore, the two numbers are:
x = sqrt(34)
y = 3*sqrt(3)