The problem states that:

1. The sum of two numbers is 65.
2. The smaller number is 17 less than the larger number.

We can set up the following system of equations based on this:

Let:

• $x$ be the larger number.
• $y$ be the smaller number.

From the problem, we have:

1. $x + y = 65$ (the sum of the numbers is 65).
2. $y = x - 17$ (the smaller number is 17 less than the larger number).

Substituting equation 2 into equation 1:

$x + (x - 17) = 65$ $2x - 17 = 65$ $2x = 82$ $x = 41$

Now, substitute $x = 41$ into equation 2: $y = 41 - 17 = 24$

Thus, the larger number is 41, and the smaller number is 24.

Would you like more details or have any questions?

Here are five related questions:

1. How would the solution change if the sum was different?
2. What if the difference between the two numbers was not 17 but a different value?
3. Can you form similar equations for different number relationships?
4. How would you verify your solution in this case?
5. What are other types of systems of equations?

Tip: When solving a system of equations, always try substituting one equation into another to reduce the number of variables!