How do you find the least common denominator of 2/(7x-14 and x/(3x-6?

May 29, 2018

L C M of $\left(7 x - 14\right) , \left(3 x - 6\right)$ is color(crimson)(=> 21 (x-2)

Explanation:

To find L C M of $\frac{2}{7 x - 14} , \frac{x}{3 x - 6}$

 7x - 14 = 7 * color(green)((x - 2)

3x - 6 = 3 * color(green)((x - 2)

L C M of $\left(7 x - 14\right) , \left(3 x - 6\right)$ is $7 \cdot 3 \cdot \left(x - 2\right)$

color(crimson)(=> 21 (x-2)

May 29, 2018

$21 \left(x - 2\right)$

Explanation:

Given: $\frac{2}{7 x - 14} \mathmr{and} \frac{x}{3 x - 6}$

Notice that both 7 from $7 x$ and 3 from $3 x$ are prime numbers. As these are the $x$ terms we will need to focus on those and just see what happens with the constants

I imagine that that the school would expect you to do it like this:

$\textcolor{w h i t e}{\text{dddddddddddddddd}} \underbrace{3 \times 7}$
Multiples of 7 $\to \textcolor{g r e e n}{7 , 14 , \textcolor{red}{21} , 28 , \text{ etc}}$

Multiples of 3 $\to \textcolor{g r e e n}{3 , 6 , 9 , 12 , 15 , 18 , \textcolor{red}{21} , 24 , \text{ etc}}$
$\textcolor{w h i t e}{\text{ddddddddddddddddddddddddd}} \overbrace{7 \times 3}$

Multiply by 1 and you do not change the value. However, 1 comes in many forms.

$\textcolor{g r e e n}{\left[\frac{2}{7 x - 14} \textcolor{red}{\times 1}\right] \mathmr{and} \left[\frac{x}{3 x - 6} \textcolor{red}{\times 1}\right]}$

$\textcolor{g r e e n}{\left[\frac{2}{7 x - 14} \textcolor{red}{\times \frac{3}{3}}\right] \mathmr{and} \left[\frac{x}{3 x - 6} \textcolor{red}{\times \frac{7}{7}}\right]}$

$\textcolor{w h i t e}{\text{d")color(green)([6/(21x-42)]color(white)("dd")and color(white)("dd}} \left[\frac{7 x}{21 x - 42}\right]$

Consider just the denominator $21 x - 42$

Notice that $2 \times 21 = 42$ so we can write it as $21 \left(x - 2\right)$