Question

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

Answer 1

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

Explanation:

`x/2=4/7`
Rearranging, `x=2(4/7)`
`x=8/7`

Answer 2

This is why cross multiply works

`x=8/7`

Explanation:

`color(blue)("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.

`color(brown)("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.
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~
`color(blue)("Answering the question using first principles")`

Given: `x/2=4/7`

We need to 'get rid' of the `2 " from " x/2`

Multiply both sides by `color(red)(2)`

`color(green)(x/2=4/7 color(white)("dddd")->color(white)("dddd")x/2color(red)(xx2)=4/7color(red)(xx2) )`

`color(white)()`

`color(green)(color(white)("dddddddddd")->color(white)("dddd.")x xx ubrace(color(red)(2)/2) =(4xx2)/7)`
`color(white)("ddddddddddddddddddddd.")darr`
But `2/2` is the same as 1 and `color(white)(".")obrace(1)xxx` gives just `x`

`color(green)(color(white)("dddddddddd")->color(white)("dddd.")x =8/7)`