The sum of the squares of two positive integers is 100 and the difference of their squares is 28. The sum of the numbers is

Concept & Formula Use basic algebraic operations to solve simultaneous linear equations formed by the squares of variables. $$ x^2 + y^2 = a $$ $$ x^2 - y^2 = b $$ Step-by-Step Solution * **Given:** Let the two positive integers be $x$ and $y$. The sum of their squares is 100, and the difference is 28. * **Calculation:** $x^2 + y^2 = 100$ (Equation 1) $x^2 - y^2 = 28$ (Equation 2) * Add Equation 1 and Equation 2 to eliminate $y^2$: $2x^2 = 128$ $x^2 = 64$ * Since $x$ is a positive integer, take the positive square root: $x = 8$ * Substitute $x^2 = 64$ into Equation 1 to find $y$: $64 + y^2 = 100$ $y^2 = 36$ $y = 6$ * Calculate the required sum of the numbers: $x + y = 8 + 6 = 14$ Exam Strategy & Shortcut **Simultaneous Addition/Subtraction:** Instead of full substitution, immediately add the two constants and halve the result to get $x^2$, then subtract them and halve the result to get $y^2$. $x^2 = (100 + 28) / 2 = 64 \Rightarrow x = 8$ $y^2 = (100 - 28) / 2 = 36 \Rightarrow y = 6$ Sum is $8 + 6 = 14$. This skips writing out the formal equations. Common Pitfall A frequent mistake is finding the values of $x^2$ and $y^2$ (64 and 36) but forgetting to take the square root before adding them, leading to an incorrect sum. Always re-read what the question is asking for at the final step. Final Answer **Therefore, the correct answer is 14.**