# How do you solve rational equations x/2 = 4/7?

Mar 3, 2018

Rearrange the given equation so that we have the unknown(s) on one side and the constants on the other.

#### Explanation:

$\frac{x}{2} = \frac{4}{7}$
Rearranging, $x = 2 \left(\frac{4}{7}\right)$
$x = \frac{8}{7}$

Mar 3, 2018

This is why cross multiply works

$x = \frac{8}{7}$

#### Explanation:

$\textcolor{b l u e}{\text{Preamble}}$

You are very likely to have been shown this in school:

This is the cross multiply shortcut approach and I would recommend you use it as it is quite fast.

$\textcolor{b r o w n}{\text{The shortcut approach is just remembering the result from first principles}}$

To 'get rid' of something that is multiply or divide turn it into 1

To 'get rid' of something that is add or subtract turn it into 0.

In doing this it turns up on the other side of the equals sign.
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$\textcolor{b l u e}{\text{Answering the question using first principles}}$

Given: $\frac{x}{2} = \frac{4}{7}$

We need to 'get rid' of the $2 \text{ from } \frac{x}{2}$

Multiply both sides by $\textcolor{red}{2}$

$\textcolor{g r e e n}{\frac{x}{2} = \frac{4}{7} \textcolor{w h i t e}{\text{dddd")->color(white)("dddd}} \frac{x}{2} \textcolor{red}{\times 2} = \frac{4}{7} \textcolor{red}{\times 2}}$

$\textcolor{w h i t e}{}$

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{dddddddddd")->color(white)("dddd.}} x \times \underbrace{\frac{\textcolor{red}{2}}{2}} = \frac{4 \times 2}{7}}$
$\textcolor{w h i t e}{\text{ddddddddddddddddddddd.}} \downarrow$
But $\frac{2}{2}$ is the same as 1 and $\textcolor{w h i t e}{\text{.}} \overbrace{1} \times x$ gives just $x$

$\textcolor{g r e e n}{\textcolor{w h i t e}{\text{dddddddddd")->color(white)("dddd.}} x = \frac{8}{7}}$