Solve:`(6x+7)/(3x+2) = (4x+5)/(2x+3)`.

Options

A. `13/9`
B. `-13/9`
C. `-11/9`
D. `11/9`

Answer

To solve the equation \(\frac{6x + 7}{3x + 2} = \frac{4x + 5}{2x + 3}\), we will follow these steps:

### Step 1: Cross-Multiply
We start by cross-multiplying to eliminate the fractions. This means we will multiply the numerator of the left side by the denominator of the right side and the numerator of the right side by the denominator of the left side.

\[ (6x + 7)(2x + 3) = (4x + 5)(3x + 2) \]

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

**Left Side:**
\[ (6x + 7)(2x + 3) = 6x \cdot 2x + 6x \cdot 3 + 7 \cdot 2x + 7 \cdot 3 \]
\[ = 12x^2 + 18x + 14x + 21 \]
\[ = 12x^2 + 32x + 21 \]

**Right Side:**
\[ (4x + 5)(3x + 2) = 4x \cdot 3x + 4x \cdot 2 + 5 \cdot 3x + 5 \cdot 2 \]
\[ = 12x^2 + 8x + 15x + 10 \]
\[ = 12x^2 + 23x + 10 \]

### Step 3: Set the Equation to Zero
Now, we set the equation from both sides equal to each other:

\[ 12x^2 + 32x + 21 = 12x^2 + 23x + 10 \]

Next, we will subtract \(12x^2\) from both sides:

\[ 32x + 21 = 23x + 10 \]

### Step 4: Rearrange the Equation
Now, we will move all terms involving \(x\) to one side and constant terms to the other side:

\[ 32x - 23x = 10 - 21 \]
\[ 9x = -11 \]

### Step 5: Solve for \(x\)
Finally, we divide both sides by 9 to solve for \(x\):

\[ x = \frac{-11}{9} \]

### Final Answer
The solution to the equation is:
\[ x = -\frac{11}{9} \]

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