Hello everyone.  ^.^  Unfortunately when I created a video for this lesson, I actually forgot that youtube has a 15 minute time limit for people like me, so I had to split it up into two parts. That aside... I wanted to finally start my thread on the basics of algebra.  ^.^  I'm hoping that more of you will actually use this thread, than used the crochet thread. haha!

Why do I have to learn this?

Contrary to what many people believe, algebra is not there to terrorize students or waste their time.  In fact, I would argue that math is not about memorizing formulas, crunching numbers, or solving for variables.  While it is true that these activities are common, they are not what define the study.  I invite you to think about it this way:  If you are interested in playing a sport, what do you partake in to prepare your body?  Would it be wise to jump into the sport with no prior knowledge or experience?  Would you expect to do well, go uninjured, and best everyone else?  The answers to these questions, of course, are no.  In order to be ready for a sport, you must exercise and strengthen your muscles.  Through this, you can practice and become a better player.  However, the routinely exercise certainly doesn't define the sport of football.

The study of mathematics is very similar.  While most machines and technologies would not be possible without math.... and while it -is- necessary some people perform advanced mathematics.... there is a reason it's a required study even for those not interested in mathematical fields as their career.  See, mathematics exercises the brain, much in the way push ups and pull ups exercise the body.  And in the same way physical exercises target parts of the body, math targets specific parts of the brain.  More specifically, it targets the parts of the brain responsible for critical and logical thinking, and also the thinking processes related to troubleshooting and prediction.  Those who struggle in math typically perform with lesser ability in these categories.

What's with the letters? I thought math was about numbers?

Since society wants students to be competitive in the real world, it's important that they study mathematics and develop their ability to think logically.  If the study of mathematics was limited to basic calculations, there wouldn't be much advancement individually or as a society.  Math has many levels and layers to explore, so let's start with the basics: one-step equations. Working through these requires one of the most basic forms of processing thought.  For example, what would your thinking process be if I asked you to answer the following question:

What number can I add to 7 in order to get 9?

More than likely, your mind starts to turn the question inside out, thinking about the number line.  What plus 7 would result in 9?  Without much thought, you could respond with the correct answer: 2.  Two, when added to seven, gives me 9.

This is a mouthful though, and it's quite cumbersome to deal with.  Machines don't run off such simple calculations, and more is needed. It would get rather ridiculous to ponder things in a similar fashion, when dealing with more advanced calculations, and when more unknowns are added. This is why a simpler way of writing such questions is needed.  For example, instead of saying "What number can I add to 7 in order to get 9?", I could just write "? + 7 = 9".  This conveys the same message, so why do we need variables?  Well, try considering the next problem:

? + ? = 10

This notation makes it seems as though you are adding two of the same number in order to get 10 (5 and 5).  Or perhaps they are different numbers which can add to be 10 (1 and 9, 1 and 8, etc).  The problem is that this notation simply isn't clear enough. This is exactly why we need 'letters' in algebra.  More specifically, we call these letters variables.  A variable is anything that represents an unknown.  It would be much more clear, for example, if we could see something like:

x + y = 10

x and y are very clearly different letters.  If I see an x and a y, I know I am looking for two -different- numbers that would add up to ten.  A variable can be any letter at all, so you might see equations like x + 3 = 100, y - 5 = 10, or 4 * k = 32.

So how does this help me solve more complicated problems?

Well, using variables and other mathematical symbols, we can develop a logical process for breaking down difficult tasks.  We can't attempt those more difficult tasks however, without fully understanding a simplified version of them.  Therefore, the first types of problems we will be looking at are called one-step equations.  A one-step equation is a problem which takes only one step to solve.  That doesn't sound so bad, right?

Let's take the following problem:

 x + 10 = 15

Although I know most of you can solve this without writing anything down, I want to remind you that we must first examine the simple things before we can delve into more complicated stuff.  So here's the goal:  We want some number that... when added to 10... becomes 15.  Essentially, we want to know what the value of x is.... or rather, we want to know what x equals.  The issue is that we do not currently know what x equals; instead, we know what x + 10 equals.  That +10 is hindering us from being able to see the barebones: x's value.  We need to get rid of it.  It's not as simple as erasing or scratching out the +10, though.  See, the equal sign means that everything on the left has an equal value to what's on the right; it's like a scale.

Imagine you had a scale, and on the left side were one red ball and three blue balls.  On the right hand side of the scale were ten blue balls. This scale balanced perfectly and every blue ball was the same size and same weight. However, you had a mission: you wanted to see how much the red ball weighed.  Now obviously, the red ball is not what's being weighed because there are three blue balls on the same side.  If you remove the three balls, will the scale remain balanced and equal?  The answer to this, of course, is no!

This is the same way an equation works.  If I have x + 10, but I want to isolate the x, I cannot simply remove the 10 and call it a day.

So how can we solve this problem of the unbalanced scales?  Well, let's take a look at the scenario with the red and blue balls, and let's imagine I had some extra blue balls sitting around.  If I added a blue ball to one side of the scale, that side would weigh more, and it would sink because the balance is lost.  However, assuming all these balls weigh the same amount, I could remedy the situation by adding a blue ball to the other side as well, right?  If I added 1 ball to each side, each side would still weigh the same amount, and the two sides would be balanced.  This works for just about any 'operation'.  For example, if I took a blue ball -away- from the left side, I could maintain equality by also removing a ball from the right side. 

Solving a one-step equation works the exact same way.  In the problem x + 10 = 15, I need to remove the ten from the left hand side of the scale.  Since I'd be taking the 10 away, this is subtraction.  However, if I subtract ten from the left hand side of the scale (the equal sign), I also have to subtract ten from the right hand side of the scale (equal sign).  Subtracting 10 from the 15 results in 5.

So what does this look like on paper?  It looks like this:

x + 10 = 15
        -10   -10     
x       =  5

Basically, in one-step equation, we perform the opposite operation of whatever is in the problem.  If you see subtraction, get rid of it through addition. If you see addition, get rid of it through subtraction. Multiplication is undone through division, and division is unraveled by multiplication.   For example, let's look at the following problem:

4 * r = 54

When you see this problem, you should interpret it like this:  I'm looking for a number that when multiplied by 4 gives me 54.  Or ask yourself: What times 4 gives me 54?  On paper, you would solve the problem by division, since the opposite of multiplication is division.  While you can divide by any number to the left and right hand side, only one number will be helpful.  For example, what if I chose to divide by 2?  That would look like this:

4 * r = 54
     2       2     

2r  = 27

The last line is a result of simplification.  4 r's, if split two ways, become 2 r's.  And 54 if split 2 ways becomes 27.  I've split both sides in half, so both sides are still equal. However, the problem is that I did not figure out what a single r equals. I only found out what 2 r's equal.

The only number that we can really divide by in order to get what we want, is the number that is already there: 4.  If we take 4 r's and split them between 4 people, each person would get 1 r, right?  Therefore, we really want to use the numbers already given to us, and divide by 4.

4 * r = 54
    4       4  
 
1r  = 13.5
(This is the same thing as r = 13.5.  The 1 is unnecessary to write, because if I show you a single r, it's clear there is only one r there.)

Can you sum this all up for me?

--In algebra, we have variables, which are traditionally represented through letters.  A variable is an unknown value which you want to discover.

--A variable can be any letter, but still represents just an unknown number.

--The goal is to isolate the variable on one side of the equal sign.

--To isolate a variable in a one-step equation, you must perform the opposite operation as what appears in the problem.

--Whatever you do to one side of an equation, you must always do to the other side.

--Placing a 1 in front of any variable is unnecessary. If there is a variable written down and nothing in front of it, it is understood there is one of them.  (i.e. k = 1k, v = 1v, x = 1x)



Here are some example problems. Note that in each one, I perform the opposite operation to isolate the variable.

    k - 3 = 7       h * 2 = 16         v / 10 = 200          x + 5 = 12
     +3   +3          2     2              *10  *10              -5   -5  

     k = 10          h = 8               v = 2000          x = 7

Once you get this part of algebra down, you're ready to delve into two step equations. For now however, I'd advise you practice this until you're comfortable with it.  If you have any questions about one-step equations, just post below and ask me!

---

Here are some videos basically going through all I wrote above. I apologize for the quiet recording, as I have no idea what the issue is.  Also, sorry about the squashed format.

Algebra?
Basically you learn how to take numbers apart and put them back together again like cars.
This a new study group or something?
I could use the practice.
Hit me up with an equation from time to time. This sounds like fun.
I deal with numbers all the time in the warehouse but this could might allow me to streach a little.
Been ages.
Good Luck.
Peace.

Really nice article and videos! I read the article and watched most of the video!

I'm in BC Calculus right now, but I remember when I was in Algebra and this would have been so useful back then! I hope the Pre-algebra and algebra I students find this, since it's so helpful!

It's been awhile since I continued on with the topic of algebra, and I think that might be due to the fact that I wasn't sure where to go from my last stopping point.  There are so many directions to go, and so much to explore on the topic of algebra!  However, one idea certainly stuck out more than the others for me... that is... the topic of distribution.  While I haven't made a video quite yet, I do expect to make one this weekend (along with replacing my trig video that got majorly screwed up).  Therefore, if you don't like reading, you might want to wait and check this thread later.

Everything is being Multiplied!

Alright, so remember those pesky multiplication tables you had to memorize in elementary school?  As it turns out, numbers aren't the only things that can be multiplied together.  I discussed this in my earlier article, but it wasn't an idea which I fully explored.  For example, we talked about the idea that 5*s could be interpreted as the value of s being multiplied by 5.  In other words, if someone tells you that 's' is a replacement for the number 4, then 5*s is the same as 5*4, or 20.  One thing I didn't really discuss though, is that there are multiple ways to represent the multiplication.  While I showed you 5*s, it could also be written as 5(s), 5xs, or even 5s.   '5xs' is really the worst notation, because the 'times' symbol could easily be mistaken for an x and be thought to be a variable.  '5s' is the simplest written form.  If you ever see a number RIGHT next to the variable, you should know that the two values are being multiplied.

When you see a number multiplied to a variable, that number is called a coefficient.  In the term 24x, 24 is the coefficient and x is the variable.  In -4h, -4 is the coefficient, and h is the variable.

As most of you are probably aware, math hardly keeps within such simple scenarios.  What if we were presented by the following scenario?:

"A small uprising company paints plates and then sells them in sets of 6.  They've received a call from a wedding planner who wants to order enough plates for three different events.  However, in order to paint the plates, the company needs to buy each plate (priced at $5.40), and enough packaging ($3.20 per plate).  How much money is the small plate company going to need?"

For anyone who runs a business, this should be a pretty simple problem to solve.  Each plate costs $5.40, and 6 plates must be bought for each set. Therefore, the cost for one set of plates would be 6($5.40), or $32.40.  The shipping cost for each plate is $3.20, so part of the overall cost for one set is also 6($3.20), or $19.20.  Therefore, the overall cost of ONE set is $32.40 + $19.20, or $51.60.  Finally, there are three sets being bought, or 3($51.60) making the cost for three sets to be $154.80.

Phew!  That is QUITE the mouthful, isn't it?  Surely, there must be a shorter way?  Well first, let's take out all the words from above and let me show you in one calculation, what we did.  This is really what we calculated:

3(6($5.40)+6($3.20))

I can just hear a few people asking for a shorter method.  To some, this may even look like absolute nonsense.  First, let's discuss an alternate method for a moment.  I knew that 3 sets of 6 plates were being made, so the cost of ONE plate could be multiplied by 18.  The cost of one plate would simply be $5.40 + $3.20, or $8.60.

How in the world does 3(6($5.40)+6($3.20)) equate to 18($8.60)???

Remember earlier when I showed you the various notations for multiplication?  One of the notations I showed you was 5(s). This means that I have 5 s's, or 5 times the value of s.  I know I'm being redundant... hang in there!  In other words... 5(s) means 5 times whatever is in the parenthesis.

Let's break the following statement down:  5( 2 + 1).  If I write 5 cherries, this means I have... a cherry plus a cherry plus a cherry plus a cherry plus a cherry, right?  Five times. This is the definition of multiplication. Just like 5s means s + s + s + s + s.  Five s's.  Therefore, 5 (2 + 1) means FIVE of those things in parenthesis.

5( 2 +1 )
= (2 + 1) + (2 + 1) + (2 + 1) + (2 + 1) + (2 + 1)  Right? 

Furthermore, those parenthesis are unnecessary.  Let's drop them and regroup.  Order of addition does not matter. For example, 3 + 2 is the same thing as 2 + 3, so I can write added numbers in any order I want to.

2 + 1 + 2 + 1 + 2 + 1 + 2 + 1 + 2 + 1

is the same as

1 + 1 + 1 + 1 + 1 + 2 + 2 + 2 + 2 + 2

Do you notice anything?  There are FIVE ones... and FIVE  twos.

That means... (hold on to your hats!) ... that I can rewrite it as 5(1) + 5(2).

Are they really equal?  Does 5( 1 + 2) = 5(1) + 5 (2).

Well, if I calculate 5( 1 + 2), I get 5(3) = 15. 
If I calculate 5(1) + 5(2), I get 5 + 10, which is ... 15!

So why is this useful information?  Well, this process is called distribution.  If I have a coefficient outside of a parenthesis, I should multiply EVERYTHING inside the parenthesis by that coefficient.  If you want the nerdspeak, that looks like this:

Law of Distribution:  a( b + c ) = ab + ac.

And guess what?  This law does not ONLY to numbers. Remember... a variable IS a number.. it's simply a number whose identity we do not know. In other words, instead of having 5( 1 + 2 ) and distributing, I can have something like 5 ( x + 2 ) and still distribute in the same way.  If I wanted to simplify 5 (x + 2), I'd have to multiply that 5 to EVERYTHING inside the parenthesis... to the x and also the 2.  It would look like this:

5 ( x + 2 ) = 5(x) + 5 (2).
Since I don't know the value of x, I can't do anything with it. I can, however, simplify the 5 (2).

5 ( x + 2 ) = 5(x) + 5 (2) = 5x + 10.

The biggest mistake that students make when faced with distribution, is only multiplying the coefficient to ONE item in the parentheses.  Remember... you must multiply the coefficient to EVERYTHING inside the parenthesis.

Let me give you one more example before releasing you to do a few on your own. Take the example of -3( x + y - 4 ).

I know this problem LOOKS different, but let me assure you that it is not.  There is still a coefficient outside a set of parenthesis.  That means that the -3 should be multiplied by EVERYTHING inside the parenthesis: the x, the y, and the -4.  (Signs are important!)

Therefore, your work should look something like this:
-3( x + y - 4 ) = -3(x) + -3(y) - -3(4)

-3( x) = -3x
-3(y) =-3y
-3(4) = 12

Therefore, I have -3x + -3y - -12.

Adding a negative number is the same thing as subtracting, so "+ -3y" may just be written as -3y.
Subtracting a negative number is like taking away (subtracting) a debt (a negative). This puts you in the positive, right?  So - -12 is really the same things as +12.

This leaves us with the following answer:

-3( x + y - 4 ) = -3x - 3y + 12.

Now, let's see if you can work some on your own. I've thrown in a few challenges, so we'll have to see which ones you can or can't handle. I've placed the answers in spoilers, so try not to look at them before you've completed the problems.

Try these out!

1. 2( x - 5)

Spoiler:
2 ( x- 5 ) = 2 (x) - 2 (5) = 2x - 10

2. -2 (x - y )

Spoiler:
-2 (x - y) = -2 (x) - -2(y) = -2x - -2y = -2x + 2y

3. 3(y + cat - 4)


Distributing to Save a Life... er... or at least... to Solve for a Value

Alright.. so hopefully by now, you don't yet have a headache. It's fine and dandy to understand distributing, but knowledge isn't entirely helpful unless you can APPLY that knowledge.

In the last article, we talked about solving one step equations.   They should have seemed pretty simple and didn't require a large process to complete. However, most of life's problems aren't solvable in one move, now are they?  Before we use distribution in our problems, let's discuss TWO step equations. Do you want to guess why they're called two step equations?  Here's one such problem:

3x - 4 = 11.

Let's consider WHAT this question is askig of us. It's asking us to find some number (x).  I should be able to multiply that number by 3 (3x), then subtract 4 from the result (- 4), and get 11 as an answer.  In other words, multiplication first... then subtraction.  Well, if I want to find the value of x, I have to 'unravel' the problem... to completely undo it. Instead of multiplying FIRST... I'm going to deal with the subtraction first.

Remember, in the last article we discussed doing the OPPOSITE of whatever the expression dictates.  If I want to eliminate that MINUS 4, I'm going to have to ADD it to both sides.  Then, in order to get rid of that three being MULTIPLIED, I'm going to need to divide it away.   The work looks like this:

3x - 4 = 11
    + 4   + 4
 3x   =   15

Then

3x = 15
  3     3

x = 5

There are two ways to look at these problems which might help you.  I know I haven't COVERED the acronym PEMDAS, but let me speak to those of you who know what it is for a moment.  The acronym PEMDAS tells you that you must simplify in the following order:

P - Parenthesis (or other grouping symbols like brackets, fractions, or radicals)
E - Exponents
MD - Multiplication or Division, in order of appearance from left to right
AS - Addition or Subtraction, in order of appearance from left to right

When solving a two step equation, you would work these in the OPPOSITE order... Or I guess... like ASMDEP?  XD

For the rest of you, let me put it in a really.... really basic way.  In a problem like 3x - 4 = 11, you see the x? The x is your goal. You want to strip everything away from the x until its naked and all alone.  The 3 is the c's clothes, the -4 is a friend standing nearby, and the = is the door.  First, you want rid of the friend.  Then the clothes come off.  (Keep it PG, people.)  In other words, look at everything that is on the SAME side of the equal sign as the x... then get rid of everything one at a time... starting with what is FURTHEST away from the x.

For example, if I had the equation 3 ( x - 4) + 5 = 12, I would only want to look at the items on the SAME side as the x... in other words, I only want to work with the 3 (x - 4 ) + 5.  What's furthest away from the x?  The + 5. You'd want to get rid of that first by subtracting it to the other side.  That'd leave you with 3 ( x- 4), which you know how to distribute.  Distributing would give you 3x - 12, and then the 12 would be furthest away. You'd add the 12 to both sides, and you'd have 3x.... off with the clothes!

Granted, however, the problem above is more than a two step equation. Don't fret if it's over your heads.  Let's look at some two step problems instead.

2x - 5 = 9  

5 is the furthest away from the x, so let's get rid of it by performing the OPPOSITE operation.  If it's being subtracted, we want to ADD it to both sides.

2x - 5 = 9
   + 5   +5
    2x  =  14

And now we need to un-do the multiplication by dividing 2 from both sides.

2x = 14
  2    2

x = 7

The really great news is this:  If you can understand a two step equation, you can work one that involves distribution.  Let's take a look at the following equation as an example:

-4 ( x - 1) = 16

We now know how to distribute.
-4(x) = -4x
-4(1) = -4

-4 (x - 1) = -4x - -4 = -4x + 4

Therefore, I can rewrite the equation like this:

-4x + 4 = 16

Now this is a two step equation which you should know how to work out.  It should look like this:

-4x + 4 = 16
   -4   -4
    -4x    = 12

-4x = 12
  -4    -4

x = -3

How about trying some on your own? Once again, I've provided the answers in spoilers for you.  

Can you Dance the Two Step? 

1. 7 ( x - 2 ) = 21

Spoiler:

 7 ( x - 2 ) = 21
7x - 14 = 21
    +14    +14
   7x   =   35

7x = 35
   7    7

x = 5

2. -3 ( x + 2 ) = 9

Spoiler:

-3 ( x + 1 ) = 9
-3x - 3 = 9
    +3    +3
   -3x = 12

-3x = 12
  -3   -3

x = -4

3. 5 ( x - 2 ) = 0

Spoiler:

5 ( x - 2 ) = 0
5x - 10 = 0
    +10   +10
   5x  =  10

5x = 10
   5  5

x = 2

In Conclusion

I know this was a bit rambly, but I really do try and explain in the most clear way possible. I think that sometimes, math isn't understood because students are used to memorization without clear understanding.  If I'm rambling TOO much though... don't ever be afraid to tell me.  And if you have any questions, just feel free to ask them here!  I can't fill this thread with practice problems, but if you google "Distribution practice" you should easily find some tutorials.  There are all kinds of resources online, but I highly suggest you find one that provides answers.  It won't do you much good if you work a few problems but have no idea what you did wrong.

And if you get the answer wrong?  Post your work here. I could tell you what you're doing wrong and hopefully move you along.

Good luck, and God bless! I love you all!