Question

The following is an indirect proof of the Division Property of Equality: For real numbers, a, b, and c, if a = b and c ≠ 0, then . Assume . According to the given information, a = b. By the Multiplication Property of Equality, one can multiply the same number to both sides of an equation without changing the equation. Therefore, . Through division, the c's cancel and ______. This contradicts the given information so . Which statement accurately completes the proof? a = b a ≠ b

Answer 1

The statement above establishes the following:

If:
(1) $a=b$
(2) $c \neq 0$

Then we must assume that in fact $a=b$, therefore if we multiply the same number, say c, to both sides of an equation this one is not affected, then we have:

(3) $ac=bc \rightarrow c(a-b)=0$

First answer.

Through division, the c's cancel and zero. Let's prove it:

a. If we divide (3) by c, this is canceled and the equation (3) is converted to:

$a-b=0$ All right up here

b. If we divide (3) by (a - b), this term is canceled and the equation (3) is converted to:

$c=0$ But this is a contradiction given that in the statement above $c \neq 0$ 

Second answer. Which statement accurately completes the proof?

The statement that completes the proof is necessarily: 

$a \neq b$

For example:

$a=5$
$b=4$
$c=9$

Given that:
$a \neq b$ and $c \neq 0$

Then:

$ac \neq bc \rightarrow 5\times9 \neq 4\times9 \rightarrow 45 \neq 36$