In an earlier section, we saw that the sentences
‘[beautiful math coming... please be patient]$\,2x - 3 = 0\,$’ and
‘[beautiful math coming... please be patient]$\,x = \frac{3}{2}\,$’ certainly look different,
but are ‘the same’ in a very important way: they always have the same truth.
No matter what number is chosen for
[beautiful math coming... please be patient]$\,x\,$, the sentences are true at the same time, and false at
the same time.
Here's the question to be explored in this section and the next:
How do we get from the harder equation
[beautiful math coming... please be patient]$\,2x - 3 = 0\,$ to the simpler equation
[beautiful math coming... please be patient]$\,x = \frac{3}{2}\,$?
That is, what can you do to an equation that will make it look
different, but not change its truth?
Two transforming ‘tools’ are needed to change
[beautiful math coming... please be patient]$\,2x - 3 = 0\,$
into
[beautiful math coming... please be patient]$\,x = \frac{3}{2}\,$.
One of these tools (the Addition Property of Equality) is discussed in this section;
the other (the Multiplication Property of Equality) is discussed in the following section.
Here's the way you would be told about one of the most commonly-used transforming
tools for equations,
using the language of mathematics:
Note: Here's how to read aloud that sentence:
$a = b$ | $\text{⇔}$ | $a + c = b + c$ |
$a$ equals $b$ | is equivalent to | $a+c$ equals $b+c$ |
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]$\, a + c = b + c\,$’?
ANSWER: You added
[beautiful math coming... please be patient]$\,c\,$ to both sides of the equation.
Hence the first part of the translation:
Continuing the translation:
What did you do to
‘[beautiful math coming... please be patient]$\,a + c = b + c\,$’ to transform it into
‘[beautiful math coming... please be patient]$\,a = b\,$’?
ANSWER: You subtracted $\,c\,$ from both sides of the equation.
Hence the rest of the translation:
So, here's the full translation of the Addition Property of Equality:
this is the way a math teacher might translate the Addition
Property of Equality,
to tell students what they can do:
Here, you will practice recognizing equivalent equations.
Also, you will identify what you are doing to one equation to get an equivalent equation.