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Hi everyone!
Welcome back to integralcalc.com.
Today we're going to be doing an optimization problem
and the one we'll be working on...
and the one we will be working on... we've been asked to find the point on the line
2x + y = 3
that is closest to the point (3, 2).
So, another words, we have a line
with the equation 2x + y = 3
and there's another point that's off the line, (3, 2),
and we've been asked to find the point on the line
that is closest to this other point.
So, first of all, let's
go ahead and graph
what this is going to look like just to give us
a better idea so we can visualize what we're doing.
The first thing I want to do is convert the equation
2x + y = 3 into
slope intercept form because,
in my opinion, that's the easiest way to graph something.
So, we've got the equation
2 x + y = 3,
to convert it into slope intercept form, we want to get y
on its own.
So in order to do that, we'll subtract 2x from both sides,
so we get y = -2x + 3.
And in slope intercept form, you will always have the
x term first and then the constant second,
and in this form, once we get
the equation on this from, with y on its own here
and x first on the right hand side,
what we know is that our slope
is right here, it's -2,
and our y intercept,
or the point in which the graph crosses the y axis,
is right here, it's 3.
So, if we go ahead and graph this
on an (x, y) coordinate plane,
we'll just go ahead and
try to draw out our axis here,
so, we've got our axis... let's just go on our graph 1,
2, 3 ,
1, 2, 3....
So,
the point in which our graph crosses the y axis is right here, so 3,
so we can go ahead... let's do this...
we can go ahead and plot that point here, right?
It crosses the y axis at 3, (0, 3), so,
that's right at (0, 3), we can plot that,
and then our slope is -2
which means that we're going to go...
let's grab this color here...
we're going to go down 2
and over 1,...
and then if we get another tick mark in here...
down 2 and over 1, right?
that's our... that's what a -2
slope tells us. So,
if we go ahead and
put in our points here,
right? then we can draw
roughly
the graph that connects these.
So,
there is our graph
of
y = -2x + 3 or
2x + y = 3.
So, we've got our graph,
and now let's go ahead and plot the point
(3, 2), so (3, 2) is going to be
right about here.
So, we want to find
the point on our graph that is closest to this point.
Now,
as you can imagine, hopefully this is somewhat logical
but, the
fastest way to get from our point to the line
is not by,
you know, me enduring
around the line, it's by going directly to the line, straight to the line, which means
we're going to draw a line
that is perpendicular exactly 90 degrees with our line
and goes through this point.
So I'm going to try to draw roughly
what that would look like
but it's something like this, right?,
It's going to go...
it's going to go exactly perpendicular to the graph, and I hope that's...
that's somewhat perpendicular,
but by perpendicular, we mean that the angle between this two here
is 90 degrees,
they are exactly perpendicular with each other.
And that is going to be
the fastest way to get from our point here
at (3, 2) to the graph.
So, we are going to be looking for
this point right here,
this is what we've been asked, right? find the point on the line,
so this point lies on our line,
that is closest to the point (3, 2),
so, that's the point on the line that is closest to
this other point right here,
(3, 2).
So, in order to
determine that mathematically, we need to first
find the equation on this orange dotted line right here,
right? we want to find that...
the equation of that line,
and then we will be able to find the point in which they intersect
(the purple and the orange lines).
So,
in order to find the equation of that line,
first we will determine the slope which we can do
because if the lines are perpendicular
we know how to find the slope of the orange line.
The way we do it,
notice that the slope of our purple line
is -2, right? that's the slope of our purple line,
to find the slope of the line that is exactly perpendicular,
first, what we're going to do is change that negative
sign to a positive, so instead of
-2 it's going to be +2,
but then we also have to...
we... we have to flip it upside down
so that our slope is 1/2.
right? So the slope of a perpendicular line
is going to be 1/2. So to take another example,
if you had a slope +6,
the slope of the perpendicular line
would be,
instead of +6, it would be -1/6.
So you flip it up side down and you change the sign,
so in our case, instead of -2, we get a +1/2,
so that's the slope of our perpendicular line.
So again, we're going to write slope intercept form,
so it will be y = (1/2)x + b.
So we plugged in 1/2
for our slope right here because we know the slope,
b is going to be our y intercept
so what we need to do now is plug in
a point for x and y
and solve for b so that we can get the equation on the line.
Well, remember that we happen to know
that (3, 2), the point (3, 2),
is on our orange line, it's right here, right?
we know that that point is on our line,
so we can go ahead and plug that
coordinate in for x and y.
So we'll plug in 3 for x and 2 for y,
so we'll get 2 = ...
1/2 oops...
1/2(3) + b,
and now we want to solve for b,
so we'll multiply everything by 2 to get rid of this fraction right here,
so we have 4 = 3 + b...
we'll subtract 3 from both sides and we'll get 1...
oh sorry... + 2b...
we'll get 1 = 2b, and then divide both sides by 2,
we'll get 1/2 = b.
So, we've solved for b,
so the equation of our
line, our oran... our orange dotted line that's perpendicular, will be
y = (1/2)x... that's the slope remember?
+ 1/2,
we solved for b and 1/2 is our
y intercept.
So that's the equation of our orange dotted line.
Now, in order to figure out...
let's go ahead and erase this so we've got some room...
to figure out this point here,...
let me grab...
this point right here
that where they intersect,
we're going to need to set the equations equal to each other.
We've got y = -2x + 3
and we've got y = (1/2)x + 1/2.
If both of those are equal to y
that means that they're also equal to each other
and that is the point in which the graphs intersect.
So we'll set those equal to each other and we'll get -2x
+ 3
= (1/2)x + 1/2.
Solving this equation for the x will give us the x coordinate
of that intersection point that we're looking for.
So, let's go ahead and multiply every term by 2 to get rid of those
fractions on the right hand side,
we'll get -4x + 6
= x + 1.
We will... Let's go ahead and add
4x to both sides, we'll get 6 =
5x + 1,
and then we'll subtract 1 from both sides
and get 5 = 5x.
Dividing both sides by 5 gives us x = 1,
so that is the x coordinate of our intersection point.
Now, to find the y coordinate, we can plug in
x = 1 to either equation
because the... that's where the lines intersect.
We would get the accurate y coordinate from either equation
but let's go ahead and just plug it in to our original function
just for the sake of
show here.
So, we'll plug it in to 2x + y = 3.
So we got x = 1,
we're plugging it in to our original equation,
so 2(1) + y = 3,
and we're solving for y to get the...
to get the y coordinate of that intersection point.
Subtracting 2 from both sides gives us y = 1.
So, that means that our
intersection point is at (1, 1)
because we solved for x we got 1, we solved for y we got 1.
So that means that the coordinates
of this point right here
are (1, 1)
and that is...
remember we were asked, we have always have to answer the...
the question that we were actually asked,
find the point on the line
2x + y = 3, that's our purple line,
that is closest to the point (3, 2) and we've done that,
the point (1, 1) does lie on the line
2x + y = 3... and that is the point on that line
that's closest to the other point out there,
(3, 2).
So our final answer would be
(1, 1),
that's it.
I hope that video helped you guys and I will see you in the next one.
Bye!!!