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To solve this, let’s define a two-digit number where the tens digit is \( x \) and the units digit is \( y \). The number itself can then be expressed as:

\[
10x + y
\]

The problem gives two conditions:

1. **The sum of the digits is one-seventh of the number**:
   \[
   x + y = \frac{1}{7}(10x + y)
   \]

2. **If the digits are reversed, the number decreases by 36**:
   \[
   10x + y - (10y + x) = 36
   \]

Let’s work through these equations step-by-step.

### Step 1: Simplify the First Condition
Starting with the first condition:
\[
x + y = \frac{1}{7}(10x + y)
\]

Multiply both sides by 7 to clear the fraction:
\[
7(x + y) = 10x + y
\]

Expanding and simplifying:
\[
7x + 7y = 10x + y
\]
\[
7y - y = 10x - 7x
\]
\[
6y = 3x
\]
\[
2y = x
\]

### Step 2: Simplify the Second Condition
Now, using the second condition:
\[
10x + y - (10y + x) = 36
\]

Expanding and simplifying:
\[
10x + y - 10y - x = 36
\]
\[
9x - 9y = 36
\]
\[
x - y = 4
\]

### Step 3: Solve the System of Equations
Now we have the following two equations:
1. \( x = 2y \)
2. \( x - y = 4 \)

Substitute \( x = 2y \) into the second equation:
\[
2y - y = 4
\]
\[
y = 4
\]

Substitute \( y = 4 \) back into \( x = 2y \):
\[
x = 2(4) = 8
\]

### Step 4: Form the Number
Now we know \( x = 8 \) and \( y = 4 \), so the two-digit number is:
\[
10x + y = 10(8) + 4 = 80 + 4 = 84
\]

### Conclusion
The two-digit number is **84**.

Would you like more details on any of these steps or have any questions?

---

Here are 5 related questions to explore:

1. How would the solution change if the sum of the digits was one-fifth of the number?
2. What if reversing the digits decreased the number by 45 instead of 36?
3. Can you apply a similar approach to solve for a three-digit number?
4. How does the method differ if we consider a base other than 10 (like hexadecimal)?
5. What are the implications of this approach in coding problems involving number manipulations?

**Tip:** Always start by defining unknowns in terms of variables and form equations based on given conditions for easier solving.

A two-digit number is such that the sum of its digits is one-seventh of the number. If the digits are reversed, the number will be decreased by 36. What is the number?

Solve the two-digit number problem where the sum of the digits is one-seventh of the number, and reversing the digits decreases it by 36. This solution covers key algebraic concepts and system of equations suitable for Grades 8-10.

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