Question
What is the least common denominator of the equation Three-fourths (x minus 3) minus one-half = two-thirds?
2
9
12
36
2
9
12
36
Answers
Answered by
GPT-4o mini
To find the least common denominator (LCD) for the fractions in the equation \(\frac{3}{4}(x - 3) - \frac{1}{2} = \frac{2}{3}\), we first identify the denominators: 4, 2, and 3.
Next, we will find the least common multiple (LCM) of these denominators:
1. The prime factorization of each of the denominators:
- \(4 = 2^2\)
- \(2 = 2^1\)
- \(3 = 3^1\)
2. For the LCM, we take the highest power of each prime number that appears in the factorizations:
- The highest power of \(2\) is \(2^2\),
- The highest power of \(3\) is \(3^1\).
3. Now, we multiply these together:
\[
\text{LCM} = 2^2 \times 3^1 = 4 \times 3 = 12
\]
Thus, the least common denominator of the equation is \(\boxed{12}\).
Next, we will find the least common multiple (LCM) of these denominators:
1. The prime factorization of each of the denominators:
- \(4 = 2^2\)
- \(2 = 2^1\)
- \(3 = 3^1\)
2. For the LCM, we take the highest power of each prime number that appears in the factorizations:
- The highest power of \(2\) is \(2^2\),
- The highest power of \(3\) is \(3^1\).
3. Now, we multiply these together:
\[
\text{LCM} = 2^2 \times 3^1 = 4 \times 3 = 12
\]
Thus, the least common denominator of the equation is \(\boxed{12}\).