In the previous section, we learned that adding/subtracting
the same number to/from both sides of an equation
makes the equation look different, but doesn't change its truth.
This tool is used to ‘transform’ an equation into (an equivalent) one that is
easier to work with.
A second transforming tool, the Multiplication Property of Equality,
is the subject of this section, and is stated below:
Note: Here's how to read aloud that sentence:
$a = b$ | $\text{⇔}$ | $ac = bc$ |
$a$ equals $b$ | is equivalent to | $ac$ equals $bc$ |
To a person not
trained in reading mathematics, the information contained
in this theorem is completely inaccessible.
If you don't understand
the language in which an idea is being expressed, then you can't
use the idea.
So, what is this theorem telling us that we can do?
To answer this question, you need to ask yourself:
What did you do to
‘[beautiful math coming... please be patient]$a = b\,$’ to transform it into
‘[beautiful math coming... please be patient]$ac = bc\,$’ ?
ANSWER: You multiplied both sides by
[beautiful math coming... please be patient]$\,c\,$.
Hence the first part of the translation:
Continuing the translation:
What did you do to
‘[beautiful math coming... please be patient]$ac = bc\,$’ to transform it to
‘[beautiful math coming... please be patient]$a = b\,$’ ?
ANSWER: You divided both sides by
[beautiful math coming... please be patient]$\,c\,$.
Hence the rest of the translation:
So, here's the full translation of the Multiplication Property of Equality:
this is the way a math teacher might translate the Multiplication
Property of Equality,
to tell students what they can do:
What goes wrong with multiplying or dividing by zero?
That is, why isn't
[beautiful math coming... please be patient]$\,c\,$ allowed to equal zero in the Multiplication Property of Equality?
First of all, recall that division by zero is undefined; it's nonsensical; it's just not allowed.
So zero certainly needs to be excluded when dividing.
But what about multiplying by zero?
The problem is that multiplying by zero can change the truth of
an equation:
it can take a false equation to a true equation.
To see this, consider the false equation
‘[beautiful math coming... please be patient]$\,2 = 3\,$’ .
Multiplying both sides by zero results in the new equation
‘[beautiful math coming... please be patient]$\,2\cdot 0 = 3\cdot 0\,$’ (that is,
‘[beautiful math coming... please be patient]$\,0 = 0\,$’), which is true.
$2 = 3$ | FALSE |
[beautiful math coming... please be patient]$2\cdot 0 = 3\cdot 0$ | multiply both sides by $0$ |
$0 = 0$ | TRUE |
Here, you will practice recognizing equivalent equations.
Also, you will identify what you are doing to one equation to get an equivalent equation.