# Comparison issue [closed]

I'm given: $\frac 12x -1 = \frac23x + \frac12$.

I'm aware that I should apply the rule where one either divides or multiplies, this time I'm told to multiply by 6.

$$\left(\frac 12x -1\right) \times 6 = \left(\frac23x + \frac12\right)\times6$$

$$\frac 62x -6 = \frac{12}{3}x + \frac62$$

$$3x-6=4x+3$$

$-x=9$ so: $x=-9$.

My question:

Why do we choose 6 to multiply? Where do we get 6 from?

## closed as unclear what you're asking by Jack, Shailesh, Namaste, JMP, user91500Aug 31 '17 at 8:47

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• 6 is divisible by both 2 and 3, and is the smallest such number. – Santeri Aug 30 '17 at 12:58
• We need a number divisble by all denominators. If we multiply with that number, all denominaotrs will vanish. The smallest such number is $6$ here, but the product of the denominators will always work as well (you do not need to find the smallest number). However, finding the smallest number makes the further calculation easier. This way, you can avoid adding and subtracting fractions. – Peter Aug 30 '17 at 12:58
• there is a mistake in the last step. Since $-x=9$ you get $x=-9$ not $x=9$. – gt6989b Aug 30 '17 at 12:59
• Tip : Always plug in the result to verify the equation. This costs not much time and prevents many mistakes! – Peter Aug 30 '17 at 13:10
• To really appreciate/understand why one puts a $6$ instead of other numbers there, it would be more useful to try to solve the equation on your own. – Jack Aug 30 '17 at 13:23

In my view it is better to start by putting the terms (summands) with the variable $x$ on one side and leave the rest on the other side.

$\frac 12x -1 = \frac23x + \frac12$.

First we add 1 on both sides to get rid of the -1 on the left side.

$\frac 12x = \frac23x + \frac12+1$.

Substracting $\frac23x$ on both sides

$\frac 12x - \frac23x= \frac12+1$.

Factoring out x on the LHS

$x\cdot \left(\frac 12 - \frac23 \right)= \frac32$

Now we take the Least common multiple (LCM) of number in the denominators 2 and 3 to sum up the fractions in the brackets on the LHS. $LCM(2,3)=6$. Thus we get

$x\cdot \left(\frac 36 - \frac46 \right)= \frac32$

$x\cdot \left( - \frac16 \right)= \frac32$

Multiplying both sides by $-6$

$x\cdot \underbrace{\left( - \frac16 \right)\cdot (-6)}_{=1}= \frac32\cdot (-6)$

$x=-\frac{18}{2}=-9$