`(3x+5)/(4x+2)=(3x+4)/(4x+7)`

Options

A. `27/19`
B. `-27/19`
C. `-19/27`
D. `19/27`

Answer

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

### Step 1: Cross-Multiply
We start by cross-multiplying the fractions to eliminate the denominators. This means we will multiply the numerator of the first fraction by the denominator of the second fraction and vice versa.

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

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

**Left Side:**
\[ (3x + 5)(4x + 7) = 3x \cdot 4x + 3x \cdot 7 + 5 \cdot 4x + 5 \cdot 7 \]
\[ = 12x^2 + 21x + 20x + 35 = 12x^2 + 41x + 35 \]

**Right Side:**
\[ (3x + 4)(4x + 2) = 3x \cdot 4x + 3x \cdot 2 + 4 \cdot 4x + 4 \cdot 2 \]
\[ = 12x^2 + 6x + 16x + 8 = 12x^2 + 22x + 8 \]

### Step 3: Set the Equation
Now we set the expanded forms equal to each other:

\[ 12x^2 + 41x + 35 = 12x^2 + 22x + 8 \]

### Step 4: Simplify the Equation
Subtract \(12x^2\) from both sides:

\[ 41x + 35 = 22x + 8 \]

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

\[ 41x - 22x = 8 - 35 \]
\[ 19x = -27 \]

### Step 5: Solve for \(x\)
Now, divide both sides by 19 to isolate \(x\):

\[ x = \frac{-27}{19} \]

### Final Answer
Thus, the solution to the equation is:

\[ x = -\frac{27}{19} \]

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