Contenuto principale

## Riepilogo: Tipi di equazioni lineari in due variabili

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# Writing linear equations in all forms

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## Trascrizione del video

A line passes through the points
negative 3, 6 and 6, 0. Find the equation of this line
in point slope form, slope intercept form, standard form. And the way to think about
these, these are just three different ways of writing
the same equation. So if you give me one of them,
we can manipulate it to get any of the other ones. But just so you know what these
are, point slope form, let's say the point x1, y1 are,
let's say that that is a point on the line. And when someone puts this
little subscript here, so if they just write an x, that means
we're talking about a variable that can take
on any value. If someone writes x with a
subscript 1 and a y with a subscript 1, that's like saying
a particular value x and a particular value of y,
or a particular coordinate. And you'll see that when
we do the example. But point slope form says
that, look, if I know a particular point, and if I know
the slope of the line, then putting that line in point
slope form would be y minus y1 is equal to
m times x minus x1. So, for example, and we'll do
that in this video, if the point negative 3 comma 6 is on
the line, then we'd say y minus 6 is equal to m times x
minus negative 3, so it'll end up becoming x plus 3. So this is a particular
x, and a particular y. It could be a negative
3 and 6. So that's point slope form. Slope intercept form is y is
equal to mx plus b, where once again m is the slope, b is the
y-intercept-- where does the line intersect the y-axis--
what value does y take on when x is 0? And then standard form is the
form ax plus by is equal to c, where these are just two
numbers, essentially. They really don't have
any interpretation directly on the graph. So let's do this, let's figure
out all of these forms. So the first thing we want to do
is figure out the slope. Once we figure out the slope,
then point slope form is actually very, very, very
straightforward to calculate. So, just to remind ourselves,
slope, which is equal to m, which is going to be equal to
the change in y over the change in x. Now what is the change in y? If we view this as our end
point, if we imagine that we are going from here to
that point, what is the change in y? Well, we have our end point,
which is 0, y ends up at the 0, and y was at 6. So, our finishing y point is 0,
our starting y point is 6. What was our finishing x
point, or x-coordinate? Our finishing x-coordinate
was 6. Let me make this very clear, I
don't want to confuse you. So this 0, we have that 0, that
is that 0 right there. And then we have this 6, which
was our starting y point, that is that 6 right there. And then we want our finishing x
value-- that is that 6 right there, or that 6 right there--
and we want to subtract from that our starting x value. Well, our starting x value is
that right over there, that's that negative 3. And just to make sure we know
what we're doing, this negative 3 is that negative
3, right there. I'm just saying, if we go from
that point to that point, our y went down by 6, right? We went from 6 to 0. Our y went down by 6. So we get 0 minus
6 is negative 6. That makes sense. Y went down by 6. And, if we went from that point
to that point, what happened to x? We went from negative 3 to
6, it should go up by 9. And if you calculate this, take
your 6 minus negative 3, that's the same thing as
6 plus 3, that is 9. And what is negative 6/9? Well, if you simplify it,
it is negative 2/3. You divide the numerator and
the denominator by 3. So that is our slope,
negative 2/3. So we're pretty much ready
to use point slope form. We have a point, we could pick
one of these points, I'll just go with the negative 3, 6. And we have our slope. So let's put it in
point slope form. All we have to do is we say y
minus-- now we could have taken either of these points,
I'll take this one-- so y minus the y value over here, so
y minus 6 is equal to our slope, which is negative
2/3 times x minus our x-coordinate. Well, our x-coordinate, so x
minus our x-coordinate is negative 3, x minus negative
3, and we're done. We can simplify it
a little bit. This becomes y minus 6 is equal
to negative 2/3 times x. x minus negative 3 is the
same thing as x plus 3. This is our point slope form. Now, we can literally just
algebraically manipulate this guy right here to put it into
our slope intercept form. Let's do that. So let's do slope intercept
in orange. So we have slope intercept. So what can we do here
to simplify this? Well, we can multiply out the
negative 2/3, so you get y minus 6 is equal to-- I'm just
distributing the negative 2/3-- so negative 2/3 times
x is negative 2/3 x. And then negative 2/3 times
3 is negative 2. And now to get it in slope
intercept form, we just have to add the 6 to both sides so
we get rid of it on the left-hand side, so let's
add 6 to both sides of this equation. Left-hand side of the equation,
we're just left with a y, these guys cancel out. You get a y is equal
to negative 2/3 x. Negative 2 plus 6 is plus 4. So there you have it, that is
our slope intercept form, mx plus b, that's our
y-intercept. Now the last thing we need
to do is get it into the standard form. So once again, we just have to
algebraically manipulate it so that the x's and the
y's are both on this side of the equation. So let's just add 2/3 x to both
sides of this equation. So I'll start it here. So we have y is equal to
negative 2/3 x plus 4, that's slope intercept form. Let's added 2/3 x, so
plus 2/3 x to both sides of this equation. I'm doing that so it I don't
have this 2/3 x on the right-hand side, this
negative 2/3 x. So the left-hand side of the
equation-- I scrunched it up a little bit, maybe more than I
should have-- the left-hand side of this equation is what? It is 2/3 x, because 2 over
3x, plus this y, that's my left-hand side, is equal to--
these guys cancel out-- is equal to 4. So this, by itself, we are in
standard form, this is the standard form of the equation. If we want it to look, make it
look extra clean and have no fractions here, we could
multiply both sides of this equation by 3. If we do that, what do we get? 2/3 x times 3 is just 2x. y times 3 is 3y. And then 4 times 3 is 12. These are the same equations,
I just multiplied every term by 3. If you do it to the left-hand
side, you can do to the right-hand side-- or you have to
do to the right-hand side-- and we are in standard form.