Division Rules

Question

Hi, I am looking for some clarity on when we CAN and CANNOT divide by a variable.

: Scenario 1: x^2 = 10x x^2/x = 10x/x would be incorrect. We would need to solve this as follows: x^2 - 10x = 0 x(x - 10) = 0 x=0 or x=10 Scenario 2: y = 4x y/x = 4x/x y/x = 4 Why is it that we cannot divide by x in scenario 1, but we can divide by x in scenario 2? Thank you for your help.

EMPOWERgmatRichC · Answer

Hi CoolFool,

You have to be careful when dividing by a variable (because you might inadvertently remove a potential solution from the equation that you're manipulating). As you've already noticed, in your first example - by diving both sides by X - you REMOVED a variable (and thus, you removed a solution). However, in your second example, by dividing by X, you did NOT remove it (you simply 'moved' it to the other side of the equation) - so that's mathematically acceptable.

GMAT assassins aren't born, they're made,
Rich

mvictor · Answer

the problem with the variables is that they can be either zero or negative. A negative number, raised to an even exponent, will yield a positive number. From here, the confusion might arise. Every time when you solve a question, ask yourself - can the exponent be zero or negative? Moreover, when you deal with an expression that has exponents >1, you need to realize that you might have more than one solution for the variable.

HKD1710 · Answer

Suggestion - If you attempt the question -  https://gmatclub.com/forum/if-ab-2b-4-a-b-what-is-the-value-of-b-232112.html

and read the explanation -  https://gmatclub.com/forum/if-ab-2b-4-a-b-what-is-the-value-of-b-232112.html#p1789369

then you doubt would be clear.

KarishmaB · Answer

Note that the scenario 1 is NOT INCORRECT. It is correct given certain constraints. 
Recall that you cannot have 0 in the denominator since it gives an undefined term. So if you know that x is not 0, you CAN divide both sides by x to get x = 10.
But, if it is possible that x is 0, you cannot divide by x. You must bring it to the left hand side. In that case, you get two possible values for x, 0 and 10.

Note that the same applies to scenario 2 as well. You CANNOT divide by x if you are not given that x is not 0. If there is a possibility that x could be 0, (say the case where y = 0, x = 0), then you cannot divide by x. Only if you are given that x is not 0, then you can divide both sides by x.

Division Rules

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