To eliminate the fractions in the given equation, let's consider the equation step-by-step:
[tex]\[
6 - \frac{3}{4} x + \frac{1}{3} = \frac{1}{2} x + 5
\][/tex]
We need to find a common number to multiply through by in order to get rid of all the fractions. This entails finding the least common multiple (LCM) of the denominators present in the equation.
### Step-by-step approach:
1. Identify all the denominators:
- [tex]\(\frac{3}{4}\)[/tex] has a denominator of 4.
- [tex]\(\frac{1}{3}\)[/tex] has a denominator of 3.
- [tex]\(\frac{1}{2}\)[/tex] has a denominator of 2.
2. Find the least common multiple (LCM) of 4, 3, and 2:
- The LCM of 4, 3, and 2 is the smallest number that all three denominators divide into without leaving a remainder.
- Start with multiples of the highest denominator, which is 4: 4, 8, 12...
- Check if each number is also divisible by 3 and 2.
3. Calculation of LCM:
- 4, 3, and 2 all divide evenly into 12.
- Hence, the LCM is 12.
4. Multiply every term in the equation by 12 to eliminate the fractions:
[tex]\[
12 \cdot 6 - 12 \cdot \frac{3}{4} x + 12 \cdot \frac{1}{3} = 12 \cdot \frac{1}{2} x + 12 \cdot 5
\][/tex]
By multiplying every term by 12, the fractions are eliminated, and we can solve the equation more easily.
The appropriate number to multiply each term by is [tex]\(\boxed{12}\)[/tex].
