Question

Create a word problem that would require the use of equations to solve the problem, and write an "answer key" which explains the solution to the problem.

Answer 1

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)