Solve: `1/4 x + 1/6 x = x-7`.

Options

A. `-6`
B. `6`
C. `-12`
D. `12`

Answer

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

### Step 1: Find a common denominator
The denominators in the left-hand side of the equation are 4 and 6. The least common multiple (LCM) of 4 and 6 is 12.

### Step 2: Rewrite the fractions with the common denominator
We can rewrite the equation as follows:
\[ \frac{1}{4}x = \frac{3}{12}x \quad \text{(since } \frac{1}{4} = \frac{3}{12}\text{)} \]
\[ \frac{1}{6}x = \frac{2}{12}x \quad \text{(since } \frac{1}{6} = \frac{2}{12}\text{)} \]
Thus, the equation becomes:
\[ \frac{3}{12}x + \frac{2}{12}x = x - 7 \]

### Step 3: Combine the fractions on the left-hand side
Now we can combine the fractions:
\[ \frac{3 + 2}{12}x = x - 7 \]
This simplifies to:
\[ \frac{5}{12}x = x - 7 \]

### Step 4: Eliminate the fraction
To eliminate the fraction, multiply both sides of the equation by 12:
\[ 12 \cdot \frac{5}{12}x = 12(x - 7) \]
This simplifies to:
\[ 5x = 12x - 84 \]

### Step 5: Move all terms involving \(x\) to one side
Now, we will move \(12x\) to the left side:
\[ 5x - 12x = -84 \]
This simplifies to:
\[ -7x = -84 \]

### Step 6: Solve for \(x\)
Now, divide both sides by -7:
\[ x = \frac{-84}{-7} \]
This simplifies to:
\[ x = 12 \]

### Final Answer
Thus, the solution to the equation is:
\[ \boxed{12} \]