The average of two numbers is 7 and three times the difference between them is 18 find the numbers

Let the two numbers be \( x \) and \( y \).

### Step 1: Use the given information to set up equations.
- **Average of two numbers**: The average of \( x \) and \( y \) is 7, so:
  \[
  \frac{x + y}{2} = 7
  \]
  Multiplying both sides by 2:
  \[
  x + y = 14
  \]

- **Three times the difference**: Three times the difference between them is 18, so:
  \[
  3(x - y) = 18
  \]
  Dividing both sides by 3:
  \[
  x - y = 6
  \]

### Step 2: Solve the system of equations.
We now have the following two equations:
1. \( x + y = 14 \)
2. \( x - y = 6 \)

Add these two equations to eliminate \( y \):
\[
(x + y) + (x - y) = 14 + 6
\]
\[
2x = 20
\]
\[
x = 10
\]

Now, substitute \( x = 10 \) into the equation \( x + y = 14 \):
\[
10 + y = 14
\]
\[
y = 4
\]

### Final Answer:
The two numbers are \( x = 10 \) and \( y = 4 \).

Would you like further details or have any questions?

Here are some related questions for practice:
1. If the average of two numbers is 12 and the difference is 5, what are the numbers?
2. The sum of two numbers is 30, and their difference is 10. Find the numbers.
3. The average of two numbers is 9, and three times their difference is 15. Find the numbers.
4. The difference between two numbers is 8, and their sum is 24. What are the numbers?
5. Two numbers have an average of 5, and their product is 12. What are the numbers?

**Tip**: When solving systems of equations, combining them by addition or subtraction can help eliminate one variable, simplifying the solution.

The average of two numbers is 7 and three times the difference between them is 18. Find the numbers.

Find two numbers where their average is 7, and three times their difference equals 18. This step-by-step guide covers key algebraic methods such as solving a system of linear equations, suitable for Grades 6-8.

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