To solve the equation [tex]\( 6 - \frac{3}{4} x + \frac{1}{3} = \frac{1}{2} x + 5 \)[/tex] by eliminating the fractions, we need to find a number that can be multiplied with every term in the equation so that all denominators are cancelled out. This number is known as the least common multiple (LCM) of all the denominators involved. Let's identify the denominators:
1. The denominator of [tex]\(\frac{3}{4} x\)[/tex] is 4.
2. The denominator of [tex]\(\frac{1}{3}\)[/tex] is 3.
3. The denominator of [tex]\(\frac{1}{2} x\)[/tex] is 2.
To find the LCM of these denominators (4, 3, and 2):
1. List the multiples of each number:
- Multiples of 4: 4, 8, 12, 16, 20, ...
- Multiples of 3: 3, 6, 9, 12, 15, ...
- Multiples of 2: 2, 4, 6, 8, 10, 12, ...
2. Identify the smallest common multiple:
The smallest number that appears in all three lists is 12.
Thus, the least common multiple of 4, 3, and 2 is 12. This means that by multiplying every term in the equation by 12, we can eliminate the fractions.
To summarize, the number that can be multiplied by each term of the equation [tex]\( 6 - \frac{3}{4} x + \frac{1}{3} = \frac{1}{2} x + 5 \)[/tex] to eliminate the fractions is [tex]\( 12 \)[/tex].
