a/b-b/a=(a+b)/ab. Tìm nghiệm. -9<=a<=9, -9<=b<=9

We are tasked with finding integer solutions for the equation:

\[
\frac{a}{b} - \frac{b}{a} = \frac{a + b}{ab}
\]
where \( -9 \leq a \leq 9 \) and \( -9 \leq b \leq 9 \).

### Step-by-Step Solution:

1. **Rewrite the equation:**

   Let's simplify both sides of the equation.
   \[
   \frac{a}{b} - \frac{b}{a} = \frac{a + b}{ab}
   \]
   The left-hand side (LHS) can be combined by finding a common denominator:
   \[
   \frac{a}{b} - \frac{b}{a} = \frac{a^2 - b^2}{ab}
   \]
   
2. **Substitute back into the equation:**
   
   Now, substitute this into the original equation:
   \[
   \frac{a^2 - b^2}{ab} = \frac{a + b}{ab}
   \]
   
3. **Simplify the equation:**
   
   Both sides have the same denominator, so we can cancel the denominator:
   \[
   a^2 - b^2 = a + b
   \]
   
4. **Rearrange the equation:**
   
   \[
   a^2 - b^2 - a - b = 0
   \]
   Factor this expression:
   \[
   (a - b)(a + b) - (a + b) = 0
   \]
   Factor out \( (a + b) \):
   \[
   (a + b)(a - b - 1) = 0
   \]
   
5. **Solve the factored equation:**
   
   For this product to be zero, either \( a + b = 0 \) or \( a - b - 1 = 0 \).

   - If \( a + b = 0 \), then \( a = -b \).
   - If \( a - b - 1 = 0 \), then \( a = b + 1 \).

### Solution 1: \( a = -b \)
   
Substitute \( a = -b \) into the range \( -9 \leq a \leq 9 \) and \( -9 \leq b \leq 9 \). The possible pairs are:
\[
(a, b) = (-9, 9), (-8, 8), (-7, 7), \ldots, (9, -9)
\]

### Solution 2: \( a = b + 1 \)

Substitute \( a = b + 1 \) into the range \( -9 \leq a \leq 9 \) and \( -9 \leq b \leq 9 \). The possible pairs are:
\[
(a, b) = (-8, -9), (-7, -8), (-6, -7), \ldots, (9, 8)
\]

### Final Answer:
The integer solutions for \( (a, b) \) are:

1. From \( a = -b \):
   \[
   (a, b) = (-9, 9), (-8, 8), (-7, 7), \ldots, (9, -9)
   \]
   
2. From \( a = b + 1 \):
   \[
   (a, b) = (-8, -9), (-7, -8), (-6, -7), \ldots, (9, 8)
   \]

Do you have any questions or would you like further clarification?

### Related Questions:
1. What happens if \( a \) and \( b \) are restricted to non-negative values?
2. Can the equation \( \frac{a}{b} - \frac{b}{a} = \frac{a + b}{ab} \) be solved for real numbers?
3. What would the solutions be if \( a \) and \( b \) were restricted to positive values only?
4. How does the solution change if \( -b = a + 1 \)?
5. Can this equation be solved graphically?

**Tip:** Factoring is often the most effective method when solving polynomial equations. Always check for common factors before expanding terms!

a/b - b/a = (a + b) / ab. Tìm nghiệm. -9 <= a <= 9, -9 <= b <= 9

Discover how to find integer solutions for the equation a/b - b/a = (a + b) / ab, with the constraint -9 <= a <= 9 and -9 <= b <= 9. This detailed solution includes factoring and rational expressions, suitable for students in Grades 9-12.

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