To eliminate the fractions in the equation \( \frac{x}{4} + \frac{x}{6} = \frac{16}{3} \), we should multiply the entire equation by the least common multiple (LCM) of the denominators: 4, 6, and 3.
-
The LCM of 4, 6, and 3 can be found by determining the highest power of each prime factor present in the denominators:
- The prime factorization of 4 is \( 2^2 \).
- The prime factorization of 6 is \( 2^1 \times 3^1 \).
- The prime factorization of 3 is \( 3^1 \).
Combining these, we take \( 2^2 \) (from 4) and \( 3^1 \) (from 6 or 3):
- LCM = \( 2^2 \times 3^1 = 4 \times 3 = 12 \).
-
Now, let's rewrite the equation after multiplying by 12:
\[ 12 \left( \frac{x}{4} + \frac{x}{6} \right) = 12 \left( \frac{16}{3} \right) \]
After calculation:
- For \( \frac{x}{4} \): \( 12 \cdot \frac{x}{4} = 3x \)
- For \( \frac{x}{6} \): \( 12 \cdot \frac{x}{6} = 2x \)
- For \( \frac{16}{3} \): \( 12 \cdot \frac{16}{3} = 64 \)
- Thus, the equation without fractions becomes \( 3x + 2x = 64 \).
Therefore, the equation should be multiplied by 12 to eliminate the fractions.