Saturday, July 19, 2014

A Fourth Math Lesson: Equation of a Line

Equation of a Line

y = mx + b

That’s it.  The end.

Well, okay.  There’s more to it than that.  But not much.  It’s pretty simple.

In this equation, (x,y) is any point on that line.  Any of them.  Pick any point, plug it into this equation, and it will work.  The left side will equal the right side.  Really.  It’s like magic.  Pretty cool, huh?  Well, it doesn’t look cool right now, because you don’t know what m and b are.  But we’re gonna get to that, and when we do– BLAMMO coolness will ensue.  So brace yourself.

“m” is the slope of the line.  That slope determines the way the line slopes.  If it slopes like this: / , then m is a positive number.  If it slopes like this: \, then m is a negative number.  Kinda cool, right?  See, I told ya.
“b” is the y intercept.  Huh?  What’s that?  Well, that’s where the line crosses the y axis in our graph.  If the line crosses the y axis above the origin, then it will be +b.  If it crosses the y axis below the origin, then it will be –b.  Simple as that.

So here’s an equation for a line:

y = 6x -7

We know just from looking at it, that the slope of our line is 6 and that our line crosses the y axis  below the origin at -7.  We even know from the fact that it’s a positive 6 that the line will slope this way:   / 
Now are you impressed?  Aw, come on.  Well, how about this trick:
You can see if a line passes through the origin by doing one simple thing.  If a line passes through the origin, then it contains the point (0,0) right?  Right.  Cuz that’s the origin.  So plug (0,0) into an equation for a line and see if it’s true.  If it is, then BLAMMO.  It passes through the origin.  If not, then WAH-WAHHH.  No go.

Try it on this one:

Y = 3x -8

And this one:

Y = -6x + 9

And this one:

Y = 12x -100

Now try this one:

Y = 18x

Ah ha!  See?  That one goes through the origin.  Can you see why, just by looking at it?  Of course you can.  Cuz you’re smart.  y = 18x is really y = 18x + 0 in disguise.  And in that line, 18 is the slope and 0 is the y-intercept.  Makes sense, doesn’t it?  If the line goes through the origin, then it intercepts the y axis at 0.  Duh.  You’re welcome.

Sometimes an equation for a line will look like this:

4x + 2y = 0

Do not be fooled.  Sure, it doesn’t look like our good friend y = mx + b, but all it takes is a little manipulation and it will. 

Remember how we learned how to isolate a variable?  Well, in y =  mx + b, y is the variable that’s isolated.  It’s all by itself in the room on the left.  Naked and everything.  So all we have to do is isolate y and we’ll have our old friend  Equation of a Line  back.
4x + 2y = 6
4x – 4x +2y = 6 - 4x
0+ 2y = 6  - 4x
2y = 6 – 4x
2y = 6 – 4x
 2         2
y = 3 – 2x  or y = -2x + 3
So just by looking at y = -2x + 3, we can see that the line has a slope of -2 and it crosses the y axis at 3.  Speaking of the y intercept being 3, here’s another cool trick for ya.  Say you have an equation that’s not all purty and in the form of y = mx + b.  Say you want to figure out where it crosses the y axis, but you don’t want to go to all the trouble to put it in the form y = mx + b first.  Say you’re too impatient for that.  What’s a faster way of figuring that out?  Well watch this:
4x + 2y = 6
We know that wherever this line crosses the y axis, at that point where it crosses, x is going to equal 0.  So that point is going to look like (0, y) where y will be the y intercept, right?  Right!  So let’s plug 0 in for x in our equation and see what happens!  Yeah!  Do it!  Do it!  Do it!     Ahem.
4(0) + 2y = 6
0 + 2y = 6
2y = 6
2y = 6
2      2
Y = 3.    Yep!  3 is the y-intercept.  The line crosses the y-axis at 3.  Almost as easy as Pi.  Get it?  Pi?  Bahaha!

A Third Math Lesson: Order of Operations

Order of Operations
Or, “Manipulating Equations to Get What You Want on Your Own TERMS” (Get it?  It’s a pun.)

You’ll notice that in both of the Isolating the Variable exercises, we dealt with addition and subtraction first, and then dealt with multiplication and division last.  In Math, this is called the proper Order of Operations.  Frankly, it’s not just the proper order, it’s the easiest order.  Another way of thinking about this is to say that we deal with an equation a term at a time. 
An equation is made up of terms.  What’s a term?  Well, a term is a chunk of an expression.  So let’s look at our old friend:
4x + 7y  - z = 42 
That’s an equation, made up of the expression 4x +7y – z, an equal sign, and the number 42.  In the expression 4x + 7y – z, we have three terms:  4x, 7y and –z.  We could also call it -1z, but in Math, we usually leave off the 1 in a term, just to keep it simple.  One z = z =1z.  Z.  We’ll leave it at that.
A term is not any old chunk of the expression.  For example, we can’t call 7y-z  a “term.”  That, my friend, would be called an “expression.”  Just a smaller expression than the one above.  A term is a nice, tidy chunk, all stuck together, without the glue of a “+” or a” – “ holding it together.  A term is just held together by multiplication or division.  So these things are all terms:
8x        17b      -2m3        149y2       -83x       ¼ h
Why do we care what a term is?  Well, because the first step in the process of Isolating the Variable is to isolate the term that contains the variable we want to isolate.  Let’s look at an example:
2a – 14b + 72c = 82
Let’s isolate the variable b.  In order to do that, we’ll need to start by isolating the term that contains b, or “-4b.”  We’re  gonna start by kicking all the other terms out of the room.  By that I mean you, 2a, and you, 72c.  Pack up your stuff, cuz you’re heading over to the other room.  But we can’t just put them there, because our equation won’t be true anymore.  So let’s do it right, and use math:
You first, 72c.  Get ready, cuz here you go:
2a – 14b + 72c = 82
2a – 14b +72c -72c = 82 -72c
2a – 14b + (72c -72c) = 82 -72c
2a -14b + 0 = 82 -72c
2a – 14b = 82 -72c
And now you, 2a.  You’re next:
2a – 2a – 14b = 82 -72c -2a
(2a – 2a) – 14b = 82 – 72c -2a
0 – 14b  = 82 – 72c – 2a
-14b = 82 – 72c – 2a
Ta da!  Now we’ve isolated the term with b in it, so we’re ready for the last step.  Right now we can see what b looks like wearing a negative 14 coat (or a -14 coat).  Let’s get him naked!  How do we do that?  Well, he’s a b multiplied by a -14.  To undo that, we’re going to have to divide both sides by -14 (dividing is the opposite of multiplying).  So now we have:
-14b = 82 – 72c -2a
  -14            -14
b = 82 – 72c -2a
So a quick review on Order of Operations when isolating a variable:
1.       Isolate the term containing the variable you want to isolate (or solve for).  To do this, perform the opposite operation of what got those terms there in the first place.  Do the same operation to both sides of the equation.
2.      Perform the opposite operation to what has already been done to your variable.  Again, make sure you do the same operation to both sides of the equation.
3.      Bask in the glory that is your naked variable.  You have solved the equation for your chosen variable.      Do a little dance. 

A Second Math Lesson: Isolating the Variable, Part Deux

Isolating the Variable, Part Deux
Or, “How to see a variable naked when there are other variables in the room.”
Sometimes your equation will have more than one variable in it.  It will look something like this:
4x + 7y  - z = 42  Think of the stuff on each side of the equal sign as being in different rooms.
You still want to see X naked, but now he’s not just wearing stuff, but 7 people named Y and a really negative punk named Z are standing in the way.  You’ve got some work to do.  Unfortunately, in Math, you can’t just make these Y and Z people evaporate.  But you can make them go to the other room and do things to make your equation look like this:
X = something.
And that’s what we want.  To get X alone.  Isolated.  And naked.  So let’s get down to business.
First, let’s get Z to leave the room.  Z got into the room by subtraction (our expression is “-z”), so to undo that, we’ll have to add:
4x + 7y – z = 42
4x + 7y –z + z = 42 + z
4x + 7y + 0 = 42 +z      or       4x + 7y = 42 + z
Now let’s kick out the Y peeps.  There were 7 of them added here, so let’s subtract 7 of them from both sides:
4x + 7y = 42 + z
4x + 7y -7y = 42 + z – 7y
4x + 0 = 42 + z – 7y
4x = 42 + z – 7y
Whew!  All that’s left in the room is X, wearing those pesky 4 shirts.  No matter.  We can tear them off his body.  He put them on via multiplication (4 times x), so all we have to do is divide:
4x = 42 + z – 7y
4x/4 =42 + z – 7y
X = 42 + z – 7y
Yes.  That does look like a mess.  X is kind of a mess.  But you’re forgetting– he’s naked!  That’s what we wanted.  That’s what we asked for. 
It’s important to be careful what you ask for.  Sometimes when you get it, you find out that it isn’t as attractive as you originally thought it would be.

A Little Math Lesson

My daughter isn't a big fan of math.  Yet.  (I only say that hopefully, not with any real expectation.)  I've struggled for years to help her with her math homework, not just because we're a mother-daughter combo, but also because I was taught to speak math fluently, both at school and at home, and she's only taught to speak it at home.  In Seattle Public Schools, the universal language of math is treated sort of like Latin-- a language that exists, and is very helpful when learning pretty much every subject in school, but one that few people take the time to learn.  My daughter is down with that.  I am not.  She's resentful that I force her to speak math at home, and I'm resentful that the whole world (or at least SPS) doesn't.

Anyhoo, this is all a long way of saying that I wrote up a few math lessons for her, teaching her math as well as math language, and I decided I wanted to share them with you, for edutainment purposes.  Here goes:

Isolating the Variable, Part One
Or, “How to see a variable naked.”

Say you have an equation that looks like this:
4x + 7 = 19. 
That’s all well and good, but really, it would be nicer if it looked like this:
X = something.
Because then we would know what X really looked like.  We could see X naked.  But for now, we can’t see X naked, because he’s wearing 4 shirts and 7 pairs of pants.  With 4 shirts on and 7 pairs of pants on, he’s a 19.  But what is he when he’s naked?  Well, we’ll have to take off the shirts and pants now, won’t we?  Yep.  So how do we do that?  And which should we take off first?
Well, when X got dressed this morning, he put on the 7 pairs of pants via addition.  Our equation  has “+7” in it.  So to undo that, we’ll need to do the opposite of addition– we’ll subtract the 7 pairs of pants.  But in order to keep it an equation (An equation is an expression that shows that two things are equal), we’ll need to subtract the 7 from both sides of the equal sign:
4x + 7 = 19
4x + 7 – 7 = 19 – 7
4x = 12
X then put on 4 shirts via multiplication.  In our equation, that looks like 4x, or 4 times x.  So to undo that, to take the shirts off, we have to do the opposite of multiplication, or division.  So we’ll divide both sides of our equation by 4:
4x = 12
4x/4 = 12/4
X = 3
Woo hooooo!  X is naked!  And when he’s naked, he’s a 3!  Who knew?  Well… um, we do.  He’s a 3.