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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