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

Multiplication or division by the same number on both sides of an equation does not change equality.

Step-by-step explanation:

Multiplication or division by the same number on both sides of an equation does not change equality. Remember that multiplication or division should apply to every term on either side of the equality. Enclose the side with more than one term in small brackets and then do the multiplication of division operation so that it applies to each term in the bracket. Multiplying both sides of the above equation with leads to:

Answer 2

The statement above establishes the following:

If:
(1)
`a=b`
(2)
`c \\eq 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 \\eq 0`

Second answer. Which statement accurately completes the proof?

The statement that completes the proof is necessarily:


`a \\eq b`

For example:


`a=5`

`b=4`

`c=9`

Given that:

`a \\eq b` and
`c \\eq 0`

Then:


`ac \\eq bc \rightarrow 5*9 \\eq 4*9 \rightarrow 45 \\eq 36`