The solution of `(2x+3)/(2x-1)=(3x-1)/(3x+1)`

Options

A. `1/8`
B. `-1/8`
C. `8/3`
D. `-8/3`

Answer

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

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

### Step 2: Expand Both Sides
Next, we expand both sides of the equation.
- Left side:
\[ (2x + 3)(3x + 1) = 2x \cdot 3x + 2x \cdot 1 + 3 \cdot 3x + 3 \cdot 1 = 6x^2 + 2x + 9x + 3 = 6x^2 + 11x + 3 \]
- Right side:
\[ (3x - 1)(2x - 1) = 3x \cdot 2x + 3x \cdot (-1) + (-1) \cdot 2x + (-1) \cdot (-1) = 6x^2 - 3x - 2x + 1 = 6x^2 - 5x + 1 \]

### Step 3: Set the Equation
Now we set the expanded forms equal to each other:
\[ 6x^2 + 11x + 3 = 6x^2 - 5x + 1 \]

### Step 4: Simplify the Equation
Subtract \(6x^2\) from both sides:
\[ 11x + 3 = -5x + 1 \]

### Step 5: Combine Like Terms
Now, we will move all terms involving \(x\) to one side and constant terms to the other side:
\[ 11x + 5x = 1 - 3 \]
This simplifies to:
\[ 16x = -2 \]

### Step 6: Solve for \(x\)
Now, divide both sides by 16:
\[ x = \frac{-2}{16} = \frac{-1}{8} \]

### Conclusion
The solution to the equation is:
\[ x = -\frac{1}{8} \]