The solution of `(x + a)/(x - 2a) = (x + 3a)/(x - 4a)` is

Options

A. `x = a`
B. `x = -a//2`
C. `x = a//2`
D. `x = -a`

Answer

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

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

### Step 2: Expand Both Sides
Now, we will expand both sides of the equation:
- Left Side:
\[ x^2 - 4ax + ax - 4a^2 = x^2 - 4ax + ax - 4a^2 = x^2 - 3ax - 4a^2 \]
- Right Side:
\[ x^2 - 2ax + 3ax - 6a^2 = x^2 + ax - 6a^2 \]

### Step 3: Set the Equation
Now we set both expanded sides equal to each other:
\[ x^2 - 3ax - 4a^2 = x^2 + ax - 6a^2 \]

### Step 4: Cancel \(x^2\) Terms
We can cancel \(x^2\) from both sides:
\[ -3ax - 4a^2 = ax - 6a^2 \]

### Step 5: Rearrange the Equation
Next, we will rearrange the equation to isolate terms involving \(x\):
\[ -3ax - ax = -6a^2 + 4a^2 \]
\[ -4ax = -2a^2 \]

### Step 6: Solve for \(x\)
Now, we can solve for \(x\):
\[ 4ax = 2a^2 \]
Dividing both sides by \(4a\) (assuming \(a \neq 0\)):
\[ x = \frac{2a^2}{4a} = \frac{a}{2} \]

### Final Answer
Thus, the solution for \(x\) is:
\[ \boxed{\frac{a}{2}} \]