# Equation solving, why does this simplification work?

$$\dfrac{154}{3.2-x} = \dfrac{66}{1.6-x}$$

How can that equation be simplified as such? :

$$154(1.6-x) = 66(3.2-x)$$

Can someone explain?

• Multiply both sides by $(3.2-x)(1.6-x)$. – J.R. Feb 28 '14 at 10:08
• ...and simplify (assuming that $x$ is not equal to $1.6$ or $3.2$ - do not forget to check this when you will have the solution of your equation). – Claude Leibovici Feb 28 '14 at 10:11
• Yes, I kind of understand that. I'm just looking for a slightly more elaborate explanation as to why it works. – James Feb 28 '14 at 10:11
• $a$ $slightly$ $more$ $elaborate$ $explanation$ ? – Claude Leibovici Feb 28 '14 at 10:12
• Sorry, you posted one second before me, I was replying to TooOldForMath. OK, that makes sense, buy why does it work? – James Feb 28 '14 at 10:15

Apart from the fact that, as long as denominators are not zero, the formal definition of fractions is that the two equations are equivalent, we have: \begin{align} \frac{154}{3.2-x} &= \frac{66}{1.6-x}\\ \frac{154}{3.2-x}\cdot (1.6 - x)& = \frac{66}{1.6-x}\cdot (1.6 - x)\\ \frac{154}{3.2-x}\cdot (1.6 - x) &= \frac{66}{\color{red}{(1.6-x)}}\color{red}{\cdot (1.6 - x)}\\ \frac{154}{3.2-x}\cdot (1.6 - x) &= 66\\ \frac{154}{3.2-x}\cdot (1.6 - x)\cdot(3.2-x) &= 66\cdot(3.2 - x)\\ \frac{154}{\color{red}{(3.2-x)}}\cdot (1.6 - x)\cdot\color{red}{(3.2-x)} &= 66\cdot(3.2 - x)\\ 154\cdot (1.6 - x) &= 66\cdot(3.2 - x) \end{align} where all the red parts cancel eachother.

• Thank you, makes sense now! – James Feb 28 '14 at 10:31

The simplified equation is reached by two performing two steps. Firstly we multiply both sides of the equation by $(3.2 - x)$:

$$(3.2 - x) * \frac{154}{3.2 - x} = \frac{66}{1.6 - x} * (3.2 - x) \\ \frac{3.2 - x}{3.2 - x} * 154 = \frac{66}{1.6 - x} * (3.2 - x) \\ 154 = \frac{66}{1.6 - x} * (3.2 - x)$$

We got rid of both of the $(3.2 - x)$'s on the left hand side since $\frac{3.2 - x}{3.2 - x} = 1$ (any number divided by itself is one e.g. $\frac{3}{3} = 1$ with the exception of $0$ as $\frac{0}{0}$ is undefined). Now repeat this whole sequence except with the $(1.6 - x)$:

$$(1.6 - x) * 154 = \frac{66}{1.6 - x} * (3.2 - x) * (1.6 - x) \\ (1.6 - x) * 154 = 66 * (3.2 - x) * \frac{1.6 - x}{1.6 - x} \\ 154 * (1.6 - x) = 66 * (3.2 - x)$$

Note that you could have multiplied by both sides by $(1.6 - x)$ as the first step then $(3.2 - x)$ as the second step and you would have arrived at the same result.