Let's solve this step by step.

We need to find two numbers, say $x$ and $y$, where $x$ is the lesser number and $y$ is the greater number. According to the problem, we have two conditions:

1. The sum of the two numbers is 65:
   $x + y = 65$

2. Twice the lesser number is 10 more than the greater number:
   $2x = y + 10$

Now, let's solve these equations simultaneously.

Step 1: Express $y$ from the first equation

From the first equation $x + y = 65$, we can express $y$ as: $y = 65 - x$

Step 2: Substitute $y$ into the second equation

Substitute $y = 65 - x$ into the second equation $2x = y + 10$: $2x = (65 - x) + 10$

Step 3: Simplify and solve for $x$

Now, simplify the equation: $2x = 65 - x + 10$ $2x + x = 75$ $3x = 75$ $x = \frac{75}{3} = 25$

So, the lesser number $x$ is 25.

Step 4: Find $y$ using the value of $x$

Substitute $x = 25$ back into the equation $y = 65 - x$: $y = 65 - 25 = 40$

So, the greater number $y$ is 40.

Final Answer

The two numbers are 25 and 40.

Would you like more details or have any other questions?

Here are five related questions:

1. How do you solve simultaneous linear equations?
2. What are the methods to solve quadratic equations?
3. How do you check if a solution to a system of equations is correct?
4. What are some examples of word problems involving systems of equations?
5. How can linear equations be applied in real-life situations?

Tip: Always double-check your solution by substituting the numbers back into the original equations.