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

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 will use cross multiplication. Here are the steps:

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

### Step 2: Expand Both Sides
Now, we will 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
Next, we will simplify the equation by subtracting \(6x^2\) from both sides:
\[ 11x + 3 = -5x + 1 \]

### Step 5: Move Terms Involving \(x\) to One Side
Add \(5x\) to both sides:
\[ 11x + 5x + 3 = 1 \]
\[ 16x + 3 = 1 \]

### Step 6: Move Constant Terms to the Other Side
Subtract \(3\) from both sides:
\[ 16x = 1 - 3 \]
\[ 16x = -2 \]

### Step 7: Solve for \(x\)
Finally, divide both sides by \(16\):
\[ x = \frac{-2}{16} = \frac{-1}{8} \]

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

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