Solve : `(2)/(3)(n+6) - (1)/(5)(n-4) = (3)/(7)(n+12)` 

Options

A. `-9`
B. 8
C. `3(1)/(9)`
D. 9

Answer

To solve the equation \(\frac{2}{3}(n + 6) - \frac{1}{5}(n - 4) = \frac{3}{7}(n + 12)\), we will follow these steps:

### Step 1: Eliminate the fractions
To eliminate the fractions, we can find the least common multiple (LCM) of the denominators (3, 5, and 7). The LCM of 3, 5, and 7 is 105. We will multiply every term in the equation by 105.

\[ 105 \left( \frac{2}{3}(n + 6) \right) - 105 \left( \frac{1}{5}(n - 4) \right) = 105 \left( \frac{3}{7}(n + 12) \right) \]

### Step 2: Simplify each term
Now, we simplify each term:

\[ 105 \cdot \frac{2}{3}(n + 6) = 70(n + 6) \]
\[ 105 \cdot \frac{1}{5}(n - 4) = 21(n - 4) \]
\[ 105 \cdot \frac{3}{7}(n + 12) = 45(n + 12) \]

So the equation becomes:

\[ 70(n + 6) - 21(n - 4) = 45(n + 12) \]

### Step 3: Distribute
Next, we distribute the terms:

\[ 70n + 420 - 21n + 84 = 45n + 540 \]

### Step 4: Combine like terms
Now, we combine the like terms on the left side:

\[ (70n - 21n) + (420 + 84) = 45n + 540 \]
\[ 49n + 504 = 45n + 540 \]

### Step 5: Isolate the variable
Next, we isolate \(n\) by moving all terms involving \(n\) to one side and constant terms to the other side:

\[ 49n - 45n = 540 - 504 \]
\[ 4n = 36 \]

### Step 6: Solve for \(n\)
Finally, we divide both sides by 4 to solve for \(n\):

\[ n = \frac{36}{4} = 9 \]

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

\[ n = 9 \]

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