One number is 7 more than the other. Twice the larger is equal to four times the smaller decreased by 2. Find the numbers.   understand the problem 

To understand the problem, let's define the two numbers step by step:

1. **Let one number be \( x \)** (the smaller number).
2. **The other number is 7 more than the smaller number**, so the larger number will be \( x + 7 \).
3. **Twice the larger number is equal to four times the smaller number decreased by 2**. Mathematically, this means:
   \[
   2(x + 7) = 4x - 2
   \]

### Steps to solve:

1. Expand both sides of the equation:
   \[
   2(x + 7) = 2x + 14
   \]
   The equation becomes:
   \[
   2x + 14 = 4x - 2
   \]

2. Move all the \( x \)-terms to one side and the constants to the other side:
   \[
   14 + 2 = 4x - 2x
   \]
   Simplifying this gives:
   \[
   16 = 2x
   \]

3. Solve for \( x \):
   \[
   x = \frac{16}{2} = 8
   \]

4. Now that we know \( x = 8 \), the larger number is \( x + 7 = 8 + 7 = 15 \).

### Final Answer:
- The smaller number is \( 8 \).
- The larger number is \( 15 \).

Would you like a detailed explanation of any step? Here are 5 relative questions to explore this concept further:

1. How would the problem change if the larger number were 10 more than the smaller one?
2. What would happen if "three times the smaller" was compared instead of "four times"?
3. How can we apply similar methods to solve problems with three unknown numbers?
4. How does this relate to solving systems of equations algebraically?
5. Can you solve this problem using a graphical approach?

**Tip:** Always label unknowns clearly and translate the word problem into equations step by step for clarity.

One number is 7 more than the other. Twice the larger is equal to four times the smaller decreased by 2. Find the numbers.

Understand how to solve a system of linear equations where one number is 7 more than the other, and twice the larger equals four times the smaller number decreased by 2. This problem uses key algebraic concepts and is suitable for students in Grades 6-8.

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