Let the two numbers be $x$ and $y$, where $x$ is the greater number and $y$ is the smaller number. We are given the following information:

1. The sum of the two numbers is 18: $x + y = 18 \quad \text{(Equation 1)}$
2. The sum of the greater number and twice the smaller number is 25: $x + 2y = 25 \quad \text{(Equation 2)}$

Solving the system of equations:

From Equation 1, solve for $x$: $x = 18 - y$

Substitute this expression for $x$ into Equation 2: $(18 - y) + 2y = 25$

Simplify and solve for $y$: $18 - y + 2y = 25$ $18 + y = 25$ $y = 25 - 18$ $y = 7$

Now substitute $y = 7$ back into Equation 1 to find $x$: $x + 7 = 18$ $x = 18 - 7$ $x = 11$

Final Answer:

The greater number is $x = 11$ and the smaller number is $y = 7$.

Would you like more details or have any questions?

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Here are 5 related questions:

1. How would the solution change if the sum of the numbers were 20?
2. What if the sum of the greater number and twice the smaller was 30?
3. How would the solution change if the sum of the two numbers was 18, but the sum of the greater and three times the smaller was given?
4. Can you solve the system using a different method, such as substitution or elimination?
5. What if the two numbers had to be consecutive integers?

Tip: When solving systems of equations, substitution is a powerful method if one variable is easily isolated.