If AB=AC, does B=C?

This is part of a series on common misconceptions.

True or False?

For real numbers $a, b,$ and $c,$ if $ab=ac,$ then $b=c.$

Why some people say it's true: Just divide both sides of the equation by $a$.

Why some people say it's false: We cannot simply divide by $a$, right?

The statement is $\color{#D61F06}{\textbf{false}}$.

We can prove that the statement is false using a simple counter-example:

Proof: Consider $a = 0, b = 1, c = 2$. We have $ab = 0 \times 1 = 0$ and $ac = 0 \times 2 = 0,$ so they are equal. However, $b \neq c$.

More generally, by the zero product property, we know that $ab =ac \implies a (b-c) = 0 \implies a = 0 \text{ or } b-c = 0$. Always remember that if we want to "divide both sides by $a$," we have to check that we are not dividing by 0.

Rebuttal: But the statement is true if $a = 1$. Then $ab = ac$ gives us $1 \times b = 1 \times c$.

Reply: Yes, the statement is true when $a = 1$. However, the question is for all possible values of $a$, $b$ and $c$. In particular, the statement need not be true when $a = 0$.

Rebuttal: But by the rules of algebra, you can divide by $a.$

Reply: Yes, but that implies that $a$ is not equal to 0. After all, you cannot divide by zero.

Want to make sure you've got this concept down? Try this problem:

See Also

• Zero Product Property
• List of Common Misconceptions