How do you solve for x: #x/(x-3) = 3/(x-3) + 3 #?

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Answer 1 · 2016-08-04T11:37:27

#x in O/#

The first thing to do here is get rid of the denominators. Note that a possible solution to the original equation must satisfy the condition

#color(purple)(|bar(ul(color(white)(a/a)color(black)(x - 3 != 0 implies x != 3)color(white)(a/a)|)))#

So, to get rid of the denominators, multiply the last term by #1 = (x-3)/(x-3)#. This will get you

#x/(x-3) = 3/(x-3) + 3 * (x-3)/(x-3)#

You can now focus exclusively on the numerators

#x = 3 + 3 * (x-3)#

Rearrange to get #x# alone on one side of the equation

#x = 3 + 3x - 9#

#x = 3x - 6#

#2x = 6 implies x = 6/2 = 3#

Notice that #x# came out to be equal to the one value that it cannot take, i.e. #x=3#.

This means that your original equation has no solution, #x in O/#.