Answer :

Final answer:

The two positive numbers in question are 7 and 12. This was determined by setting up a system of two equations based on the given problem and then solving for x using algebraic methods.

Explanation:

In mathematics, specifically in algebra, we can solve this problem by setting up a system of two equations. Let's call the smaller number x, meaning the larger number is x + 5 as stated by the problem ('one positive number exceeds another by 5').

The problem also says that the sum of the squares of these numbers is 193 ('the sum of their squares is 193'). Thus we can set up our second equation as x^2 + (x + 5)^2 = 193.

To solve this system, you first expand (x + 5)^2 to get x^2 + 10x + 25 and substitute this into our second equation to get x^2 + x^2 + 10x + 25 = 193. Combine like terms to get 2x^2 + 10x + 25 = 193. Subtract 193 from both sides to set the equation to 0: 2x^2 + 10x + 25 - 193 = 0, which simplifies to 2x^2 + 10x - 168 = 0.

At the end, you divide all terms by 2 to simplify and get x^2 + 5x - 84 = 0. Solving this quadratic equation either by factoring or using the quadratic formula, you find x = 7 or x = -12. Since we're talking about positive numbers, x = 7 is our solution, making the larger number 7 + 5 = 12. So, the two numbers are 7 and 12.

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