Questions: Find the least common denominator of 4x/(5x-15) and 7x/(3x-9)

Transcript text: Find the least common denominator of $\frac{4 x}{5 x-15}$ and $\frac{7 x}{3 x-9}$

Solution

Step 1: Identify the Denominators

The given fractions are:

\[ \frac{4x}{5x - 15} \quad \text{and} \quad \frac{7x}{3x - 9} \]

The denominators are $5x - 15$ and $3x - 9$.

Step 2: Factor the Denominators

Factor each denominator:

For $5x - 15$, factor out the greatest common factor (GCF), which is 5:

\[ 5x - 15 = 5(x - 3) \]
For $3x - 9$, factor out the GCF, which is 3:

\[ 3x - 9 = 3(x - 3) \]

Step 3: Determine the Least Common Denominator (LCD)

The least common denominator is the least common multiple (LCM) of the factored denominators. The factored forms are:

The LCM must include each factor the greatest number of times it appears in any of the factorizations:

Thus, the LCM is:

\[ \text{LCM} = 5 \cdot 3 \cdot (x - 3) = 15(x - 3) \]

The least common denominator of $\frac{4x}{5x - 15}$ and $\frac{7x}{3x - 9}$ is $\boxed{15(x - 3)}$.

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