Let the two numbers be $x$ (the greater number) and $y$ (the lesser number).

Step 1: Form the equations

From the problem, we know:

1. The sum of the two numbers is 18: $x + y = 18$
2. The sum of the greater number and twice the lesser number is 25: $x + 2y = 25$

Step 2: Solve the system of equations

We have the system of equations: $x + y = 18 \quad \text{(Equation 1)}$ $x + 2y = 25 \quad \text{(Equation 2)}$

Subtract Equation 1 from Equation 2 to eliminate $x$: $(x + 2y) - (x + y) = 25 - 18$ $x + 2y - x - y = 7$ $y = 7$

Step 3: Substitute $y = 7$ into one of the equations

Substitute $y = 7$ into Equation 1: $x + 7 = 18$ $x = 18 - 7$ $x = 11$

Final Answer:

The two numbers are 11 (greater) and 7 (lesser).

Would you like more details or further explanation?

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Here are some related questions to explore:

1. What if the sum of the two numbers was 20 instead of 18?
2. Can you solve the problem using substitution or elimination methods differently?
3. What happens if both numbers were negative?
4. How would this change if you had three unknowns instead of two?
5. Can you generalize this problem for any sum of two numbers?

Tip: For systems of equations, try to eliminate one variable by either adding or subtracting equations to simplify the solving process.