Summary:

- "There is no Subtraction - I should add the opposite instead"

- "There is no Division - I should multiply by the inverse instead"

- "Algebra uses Addition and Multiplication ONLY!!"

- "There is no Subtraction - I should add the opposite instead"

- "There is no Division - I should multiply by the inverse instead"

- "Algebra uses Addition and Multiplication ONLY!!"

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## The Math Tutor - Austin

The First Rule of Algebra - Or, "No, Virginia - There Is No Subtraction!"

If I had to pick the most important lesson you could learn in Algebra, chances are it might be this one...

(Even worse - If I had to pick the lesson that's least taught, it'd also be this one - so take the opportunity to learn something special!)

Algebra has a lot to do with rearranging things

You don't have to sit in an algebra classroom very long before you find out that you're spending a lot of your time trying to puzzle out riddles about numbers. For example, the equation

3x + 2 = 58 - 4x

is really a riddle about 'x' - what number could take the place of 'x' and make this equation a true statement?

Of course, when it's written like that, who knows what 'x' will work??

BUT - if we could rearrange things to make the riddle simpler, like

x = 8,

then it would be a LOT easier to say what x will work!

(Don't worry for now about how I rearranged this - we'll get to that later. Instead, just remember that "rearranging" is important in Algebra!)

Some things "rearrange" better than others...

When you were studying arithmetic, you probably learned that there were four basic things you could do with two numbers - add, subtract, multiply, and divide.

Now, of these four operations, some are better for rearranging than others. For example, when we look at addition and subtraction:

3 + 4 can be rewritten as 4 + 3 . It gives the same answer either way.

BUT

3 - 4 CANNOT be rewritten as 4 - 3 !

You can see the same issue with multiplication and division:

3 * 4 is the same as 4 * 3

BUT

3 / 4 is NOT the same as 4 / 3

(Math teachers will tell you that it's because addition and multiplication are "commutative", and subtraction and division are not.)

If you remember anything from this, it should be that:

Addition and Multiplication can be rearranged (yay for Addition!!)

BUT

Subtraction and Division CANNOT! (boo for Subtraction!!)

If Algebra is about rearranging, how can we deal with those awful facts about Subtraction and Division?

Subtraction and Division are hereby BANISHED from Algebra!!

Yep. You read that one right. Subtraction and Division don't rearrange - so...

We're getting rid of them! Forever!

(Uh - but wait - what if I really do need to subtract something?)

Never fear, gentle reader - there's always a way...

Opposites and Inverses to the rescue!

It turns out that we don't even NEED subtraction or division if we play our cards right.

Instead of subtracting a number, we'll just add the opposite! [Keep in mind that every number has an opposite - (-4) is the opposite of 4, and 5 is the opposite of (-5), etc.]

Example:

3 - 4 can be rewritten as 3 + (-4)

3 - (-4) can be rewritten as 3 + 4

a - b can be rewritten as a + (-b)

3 - (x+2) can be rewritten as 3 + (-1)*(x+2) [ remember to write "(-1)*( ... )" rather than "-( ... )" - it works better that way... ]

Instead of dividing by a number, we'll just multiply by the inverse! [Keep in mind that (almost) every number has an inverse - (1/2) is the inverse of 2, (4/3) is the inverse of (3/4), 5 is the inverse of (1/5), etc. If you're clever, you'll see that the inverse means the same thing as "reciprocal"] [OH - and one number doesn't have an inverse. Do you know what it is?]

Example:

3 / 4 can be rewritten as 3 * (1/4)

5/(x+2) can be rewritten as 5 * 1/(x+2)

(x+3)/6 can be rewritten as (x+3) * (1/6)

< yet to come - practice problems in rewriting to remove subtraction and division... >

One Final Reminder - Memorize these and Algebra will be Smooth Sailing!

Repeat these to yourself every night as you go to sleep:

- "There is no Subtraction - I should add the opposite instead"

- "There is no Division - I should multiply by the inverse instead"

- "Algebra uses Addition and Multiplication ONLY!!"

All this may seem a bit silly at the moment - but I promise you - if you can get good at removing subtraction and division from your algebra problems, then you will save many a tear. And - even better - you'll find that Trigonometry and Calculus get a lot easier, too!