Solving for Two Numbers: Sum and Difference of Squares

Let the two positive numbers be $x$ and $y$, with $x$ being the greater number.

Defining Equations from the Problem

The problem provides two conditions:

• The sum of the two numbers is 27: $x + y = 27$
• The difference of their squares is 81: $x^2 - y^2 = 81$

Calculating the Difference Between the Numbers

Use the algebraic identity for the difference of squares: $x^2 - y^2 = (x - y)(x + y)$.

Substitute the given values:

$ 81 = (x - y)(27) $

Solve for the difference $(x - y)$:

$ x - y = \frac{81}{27} $

$ x - y = 3 $

Finding the Greater Number

Now we have a system of two linear equations:

1. $x + y = 27$
2. $x - y = 3$

To find the value of $x$, add the two equations together:

$ (x + y) + (x - y) = 27 + 3 $

$ 2x = 30 $

Divide by 2 to find $x$:

$ x = \frac{30}{2} $

$ x = 15 $

The value of the greater number ($x$) is 15.