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!

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