Question

Two numbers are each multiplied by themselves to give two new numbers, The difference between these two numbers is less than ten; the difference between the two original numbers was one. The two original numbers added together was more than 7. What is one of the original numbers?

Answer 1

let the two original numbers be \(x,y\) with \(x\) being the larger of the two. then our condition are expressed by the 3 following equations:
\[x^2-y^2<10\]\[x-y=1\]\[x+y>7\]
if you solve the second equation for (x) and substitute that into the third equation, you"ll find the lowest possible value for the smaller number. If you choose too large a value, you will find that the first equation is no longer true.

For example, if we take \(9,8\) as our two values, \(9-8=1\) so the second equation is true, and \(9+8>7\) so the third equation is true, but \(9^2-8^2 =81-64=17\) so \(9^2-8^2 <10\) is not true.