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

#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.
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#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) #