Q&A

# Missing a solution: Are A and B always equal if A-B=0

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I just came across this problem on an online learning app:

4x(5−x)−12(5−x)+100=100

I tried to solve it by subtracting 100 from each side then inferring if something minus something else equals zero then those two things must be equal.

4x(5−x)=12(5−x)

Divide both sides by (5-x) and x must be 3.

But brilliant had a different way of solving it that revealed there are two possible values for x.

Where did I make a mistake in my approach to solving the problem?

Is it incorrect to assume that if a minus b equals 0 then a must equal b?

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Dividing by 5-x is valid only if x is not 5. But x=5 is another solution to the original equation.

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One way to think about the question is that you get $$(4x-12)(5-x) = 0.$$

A product $ab$ is zero if and only if $a = 0$ or $b = 0$. So you get either $4x=12$ or $5=x$, thus getting your two solutions. No division required.

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To directly answer the question in the title: yes, $a-b=0$ always implies $a=b$.

(Well, okay, I won’t say there isn’t some abstract algebra, somewhere, where it doesn’t hold, and where $a-b+b=a$ isn’t necessarily true. But in our ordinary, everyday algebra, it’s true.)

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