If `(3 a + 4b)/( 3 c + 4 d) = (3 a - 4 b)/( 3 c - 4 d) ` then

Options

A. ab = cd
B. ad = bc
C. ac = bd
D. ` a = b = c ne d `

Answer

To solve the equation

\[ \frac{3a + 4b}{3c + 4d} = \frac{3a - 4b}{3c - 4d}, \]

we will use cross-multiplication.

### Step 1: Cross-Multiply
Cross-multiplying gives us:

\[ (3a + 4b)(3c - 4d) = (3a - 4b)(3c + 4d). \]

### Step 2: Expand Both Sides
Now, we will expand both sides of the equation.

**Left Side:**
\[ (3a + 4b)(3c - 4d) = 3a \cdot 3c - 3a \cdot 4d + 4b \cdot 3c - 4b \cdot 4d = 9ac - 12ad + 12bc - 16bd. \]

**Right Side:**
\[ (3a - 4b)(3c + 4d) = 3a \cdot 3c + 3a \cdot 4d - 4b \cdot 3c - 4b \cdot 4d = 9ac + 12ad - 12bc - 16bd. \]

### Step 3: Set the Expanded Forms Equal
Now we set the expanded forms equal to each other:

\[ 9ac - 12ad + 12bc - 16bd = 9ac + 12ad - 12bc - 16bd. \]

### Step 4: Simplify the Equation
Subtract \(9ac\) and \(-16bd\) from both sides:

\[ -12ad + 12bc = 12ad - 12bc. \]

### Step 5: Rearranging Terms
Now, let's rearrange the terms:

\[ -12ad - 12ad = -12bc - 12bc, \]

which simplifies to:

\[ -24ad = -24bc. \]

### Step 6: Divide by -24
Dividing both sides by -24 gives us:

\[ ad = bc. \]

### Conclusion
Thus, we conclude that:

\[ ad = bc. \]