Solve : `(a-2)/(a+4)=(a-3)/(a+1)`

Answer

To solve the equation \(\frac{a-2}{a+4} = \frac{a-3}{a+1}\), we will follow these steps:

### Step 1: Cross Multiply
We start by cross-multiplying the fractions to eliminate the denominators:
\[ (a - 2)(a + 1) = (a + 4)(a - 3) \]

### Step 2: Expand Both Sides
Next, we will expand both sides of the equation:
- Left side:
\[ (a - 2)(a + 1) = a^2 + a - 2a - 2 = a^2 - a - 2 \]
- Right side:
\[ (a + 4)(a - 3) = a^2 - 3a + 4a - 12 = a^2 + a - 12 \]

So, we have:
\[ a^2 - a - 2 = a^2 + a - 12 \]

### Step 3: Simplify the Equation
Now, we will simplify the equation by subtracting \(a^2\) from both sides:
\[ -a - 2 = a - 12 \]

### Step 4: Rearrange the Terms
Next, we will move all terms involving \(a\) to one side and constant terms to the other side:
\[ -a - a = -12 + 2 \]
This simplifies to:
\[ -2a = -10 \]

### Step 5: Solve for \(a\)
Now, we will divide both sides by -2 to solve for \(a\):
\[ a = \frac{-10}{-2} = 5 \]

### Final Answer
Thus, the solution to the equation is:
\[ \boxed{5} \]

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