Question

One number is 7 more than another.The difference between their squares is 217.What are the numbers?

Answer 1

19 and 12

Explanation:

Let a and b be these numbers.

1. If one number is 7 more than another, then

`a-b=7.`

2. If the difference between numbers squares is 217, then

`a^2-b^2=217.`

3. Solve the system of two equations:

`\left\{\begin{array}{l}a-b=7\\a^2-b^2=217\end{array}\right.\Rightarrow \left\{\begin{array}{l}a-b=7\\(a-b)(a+b)=217\end{array}\right.\Rightarrow \left\{\begin{array}{l}a-b=7\\a+b=(217)/(7)=31\end{array}\right..`

Add these two equations:

`a-b+a+b=7+31,\\ \\2a=38,\\ \\a=19.`

Then

`19-b=7,\\ \\b=19-7=12.`