>> Welcome to the Cypress College Math Review on finding equations of lines.
Objective one, find the equation of a line given any point and the slope.
So find the equation of a line given any point and the slope of the line.
Step one will be to call the given point X1, Y1.
Step two will be to plug-in values into point slope form,
where m represents the slope of the line.
And step three will be to solve for Y. For this example, we want to write the equation
of the line with a given point and slope.
So our given point is 3 comma 10 and our slope is 1 over 3.
So our first step is to call the given point 3 comma 10.
We want to call this X1, Y1.
So our X1 will be equal to 3 and our Y1 will be equal to 10.
Step two is to plug these values into point slope form.
So let's first write point slope form.
And then now, let's plug-in these values.
So Y we just leave as a variable minus Y1 well Y1 is equal to 10,
equals m. Remember m represents our slope.
In the problem, they tell us to use a slope of 1 over 3, then times X, again we also leave
as a variable, minus X1 while X1 is going to be equal to 3.
So step three is to solve for Y. So if I want to solve this equation for Y,
the first thing we should do is distribute the 1/3 to the X minus 3.
So let's go ahead and do that.
So we have Y minus 10 equals, now distributing the 1/3,
we get 1/3X minus the 1/3 times 3 will just multiply to become one.
And again, if we need to solve for Y, we should add 10 to both sides.
And then let's see what we're left with.
So on the left-hand side, we're just left with Y. Then equals 1/3 X.
And then minus 1 and plus 10 will give us plus 9.
For this next example, we want to write the equation
of the line with a given point and slope.
So here, I'll given point is zero comma negative 7, and we have a slope of negative 4.
So step one is to call the given point.
Zero comma negative 7.
We wish to call this X1, Y1.
So X1 will be equal to zero, and Y1 will be equal to negative 7.
Step two is to plug these values into point slope form.
So let's first write down point slope form.
Now, let's go ahead and plug in these values.
So Y we just leave as a variable.
Minus Y1 well Y1 is equal to negative 7, equals M our slope.
In the problem, they tell us to use negative 4 as our slope times X we just leave
as a variable minus X1, while X1 is just equal to zero.
Step three is to solve this equation for Y. So looking at the left-hand side,
since we have those two negatives, that would just become a plus.
So on the left, we get Y plus 7 equals.
Now, looking at the right hand side inside of the parentheses, we have X minus zero.
Well, X minus zero just becomes X. So on the right-hand side,
we're just left with negative 4X.
So to solve for Y, we should subtract both sides by a 7.
It when we do that, on the left-hand side we're just left with Y
and then equals negative 4X minus 7.
For this example, we want to write the equation of the line with a given point and slope.
So our given point is negative 2 comma negative 5.
And here we want an undefined slope.
So since we want an undefined slope, this example is going to be different
than the previous two because you can't plug-in anything in for m,
since again we want an undefined slope.
So recall that vertical lines have an undefined slope.
So keeping that in mind, I again want this line to go
through the point negative 2 comma negative 5.
And since we want an undefined slope, I know that this has to be a vertical line going
through the point negative 2 comma negative 5.
So notice that this vertical line crosses the X axis at negative 2.
So then, the equation of this line is just X equals negative 2.
Pause the video and try these problems.
Objective two.
Find the equation of a line given two points.
To find the equation of a line given two points on that line,
step one will be to find the slope using the formula for slope.
Step two will be to choose any point and plug-in values
of the point and slope into point slope form.
And step three will be to solve for Y. For this example, we want to find the equation
of the line passing through the two points negative 4 comma 5 and negative 8 comma 8.
So step one is to find the slope using the formula for slope.
So in order to use the formula for slope, we have to work with our two given points here,
the negative 4 comma 5, and the negative 8 comma 8.
And so we'll call the negative 4 comma 5 our X1, Y1.
And we'll call the negative 8 comma 8 our X2, Y2.
So let's first write the formula for slope.
So now, just plugging in our values, Y2 is equal to 8, minus Y1 is equal to 5 over X2.
X2 is equal to negative 8 minus X1, which is equal to negative 4.
So from here, all we have to do is just simplify.
So we get that m is equal to 8 minus 5.
That will give us 3 over negative 8 minus a negative 4 will just give us negative 4.
So once we have our slope, our step two is to choose any point and plug-in values
of the point and slope into point slope form.
So again, we can choose either the two points negative 4 comma 5 or negative 8 comma 8.
I'll choose the negative 4 comma 5.
And we just found our slope to be negative 3 over 4.
So what we want to do is plug these values into point slope form.
So our X1 will be negative 4.
Our Y1 will be equal to 5.
This go ahead and first write down what point slope form is.
And then now, just plugging in all of these values and we leave Y as our variable minus Y1.
Well, Y1 is equal to 5 equals m our slope that was negative 3 over 4 times X we also leave
as a variable minus X1, while negative 4 is equal to X1.
So simplifying inside of those parentheses where we see X minus a negative 4,
we can just rewrite that as X plus 4.
So then we get Y minus 5 is equal to negative 3/4 times X plus 4.
And so, step three is to solve for Y. So again,
we have the equation Y minus 5 equals negative 3/4 times X plus 4.
And all we have to do now is solve for Y. So the first thing that I'll have
to do is distribute the negative 3/4 to everything in the parentheses.
So we get Y minus 5 equals, distributing the negative 3/4 we get negative 3/4 X.
And then negative 3/4 times 4 will just leave us with a minus 3.
And solving for Y, we should add 5 to both sides.
And when we do that, we are left with Y equals negative 3/4 X,
and then negative 3 plus 5 will give us plus 2.
For this example, we want to find the equation of the line passing
through the two points 5 comma negative 3 and 2 comma negative 3.
So step one is to find the slope using the formula for slope.
So in order to do that, we need to work with our two given points,
the 5 comma negative 3 and the 2 comma negative 3.
So we'll call the 5 comma negative 3 our X1, Y1.
And we'll call the 2 comma negative 3 our X2, Y2.
So next, let's write the formula for slope.
And then from here, we just want to plug in the values.
So Y2 is equal to negative 3.
Minus Y1 is also equal to negative 3.
All over X2.
Well, X2 is equal to 2 minus X1 is equal to 5.
So now, just simplifying.
Negative 3 minus a negative 3 will just give us zero over 2 minus 5 gives us negative 3.
But if we have a zero in the numerator of a fraction,
that will just give us a slope of zero.
So then step two is to choose any point and plug-in values
of the point and slope into point slope form.
So we can choose either of the two given points.
This go ahead and choose the 5 comma negative 3.
And then we also need our slope, which we just found to be equal to zero.
So we want to plug these values into point slope form.
Let's go ahead and write point slope form.
And so, we'll call the 5 comma negative 3 our X1, Y1.
And again, our slope is zero.
So now, just plugging in the values, we get Y minus Y1 is equal
to negative 3 equals m our slope, which we found
to be zero times X minus X1, where X1 is equal to five.
So looking at the left-hand side, Y minus a negative 3, since we have those two negatives,
that just becomes Y plus 3 equals.
And then on the right-hand side, zero times anything just gives us zero.
So step three is to solve for Y. So if I want to solve this equation for Y,
all we have to do is just subtract three to both sides of the equation.
And when we do that, we are left with Y equals negative 3.
Here we want to find the equation of the line passing
through the two points negative 1 comma negative 8 and 2 comma zero.
So step one is to find the slope using the formula for slope.
So in order to use the formula for slope, we need to work with our two given points,
the negative 1 comma negative 8, and the 2 comma zero.
So our negative 1 comma negative 8, that will be our X1, Y1.
And the 2 comma zero will be our X2, Y2.
So next, let's write down the formula for slope.
So plugging in the values, Y2 is equal to zero minus Y1 is equal to negative 8 over X2.
X2 is equal to 2 minus X1, which is negative 1.
So all we have to do now is just simplify.
So zero minus a negative 8, that just becomes a positive 8
over 2 minus a negative 1 gives us a positive 3.
So we get a slope of 8 over 3.
Step two is to choose any point and plug-in values
of the point in slope into point slope form.
So again, we get to choose either of the two given points.
For this example, I'm actually going to choose the point 2 comma zero.
That point will be a little bit easier to work with, since we have a zero there.
And then we also need to use our slope, which again, we just found to be 8 over 3.
So let's write point slope form down.
And so, our 2 comma zero that is going to be our X1, Y1.
So now, plugging in the values into point slope form, we get Y minus Y1.
We're plugging in zero for Y1, equals m our slope.
We found that to be 8 over 3 times X minus X1.
For X1 we would be plugging in 2 for X1.
So now, step three is to solve for Y. So solving this equation for Y,
notice on the left-hand side we have Y minus zero,
so then that's just going to give us Y equals.
And then from here, I want to distribute that 8/3 to the X minus 2.
To distributing, we get 8/3X minus.
And then 8/3 times 2 will just give us 16 over 3.
Pause the video and try these problems.
Objective three -- Find the equation of a line given a point
and a line parallel or perpendicular to the line.
So to find the equation of a line given a point and given a line, that the line is parallel
or perpendicular to -- step one will be to determine the slope of the line.
So in order to determine the slope, we need to know that parallel lines have the same slope,
and perpendicular lines have slopes that are opposite reciprocals.
So let's talk about that for a little bit, what we mean by opposite reciprocals.
So let's say we have the number 3 over 4.
If we want to the opposite reciprocal of 3 over 4, well, if we need the opposite since 3
over 4 positive, the opposite will be negative.
And since we need the reciprocal, reciprocal just tells us to flip the fraction.
So then we would get a negative 4 over 3.
So 3 over 4 and negative 4 over 3 are what we call opposite reciprocals.
In step two will be to plug in the values of the point and the slope into point slope form.
And step three will be to solve for Y. Here we want to find the equation of the line given
that it is parallel to the line Y equals 1/5 X plus 3 and passes
through the point negative 10 comma 2.
So step one is to determine the slope of the line.
So notice in the problem it's telling us that the line must be parallel
to the line Y equals 1/5 X plus 3.
So this equation is of the form Y equals MX plus B. So then,
the slope of this line is going to be 1 over 5.
So since the line that we're trying to find needs to be parallel to this line,
I know that parallel lines have the same slope.
So the slope of the line that we need to find will also have a slope of 1 over 5.
Step two is to plug in the values of the point and the slope into point slope form.
So we know that our slope is 1 over 5,
and we also need to use the given point negative 10 comma 2.
So let's go ahead and write point slope form down.
So then, our given point the negative 10 comma 2, that will be our X1, Y1.
So now, plugging in the values into point slope form, we get Y minus, Y1 is equal to 2,
equals M our slope, is 1 over 5 times X minus X1, where X1 is equal to negative 10.
So let's simplify this equation a bit.
So looking inside of the parentheses on the right-hand side, we have those two negatives.
So those two negatives will become a positive.
So rewriting this a bit, we get Y minus 2 equals 1/5 X plus 10.
So then, step three is to solve this equation for Y. So let me rewrite the equation up here.
So if we wish to solve this equation for Y,
the first thing that we should do is distribute the 1/5 to the X plus 10.
So to do that, we get Y minus 2 equals distributing the 1/5.
We get 1/5 X plus.
Then 1/5 times 10 will just give us 2.
So if we need to solve this for Y, we just have to add 2 to both sides of this equation.
Adding 2 to both sides of this equation, we are left with Y equals 1/5 X.
And then 2 plus 2 gives us plus 4.
For this example, we want to find the equation of the line given that it is perpendicular
to Y equals negative 2/3 X minus 7, and passes through the point negative 4, comma zero.
So step one is to determine the slope of the line.
So in the problem, it tells us that our line that we need to find must be perpendicular
to the line Y equals negative 2/3 X minus 7.
So notice that this equation is of the form Y equals MX plus B. So I know that the slope
of this line is going to be negative 2 over 3.
So since the line that we're trying to find must be perpendicular to this line,
the slope that we need to use will have to be the opposite reciprocal of negative 2 over 3.
So since that has to be the opposite of negative 2 over 3,
since negative 2 over 3 is negative, the opposite should be positive.
So I know our slope will be positive and since we need the reciprocal of 2 over 3,
that just means we need to flip that fraction.
So then, the slope that we should be using will be a positive 3 over 2.
Step two is to plug in the values of the point and the slope into point slope form.
So we have our slope of 3 over 2, and we also need
to use our given point of negative 4 comma zero.
So writing point slope form down --
We will be using our given point negative 4 comma zero as our X1, Y1.
So now, plugging in the values, we get Y minus Y1, but we're plugging
in zero for Y1, equals M our slope.
That's 3 over 2 times X minus X1 and we're plugging in negative 4 for X1.
So simplifying equation a bit, looking at the left-hand side,
Y minus zero, that's just to give me Y equals.
And then, looking inside of the parentheses since we have two negatives,
that's just going to change to a plus.
So on the right-hand side, we get 3 halves times X plus 4.
So step three is to solve for Y. So solving this equation for Y,
let me bring the equation that we're working with up here.
And if I need to solve this equation for Y,
all we have to do is just distribute that 3 over 2 to the X plus 4.
So distributing the 3 over 2, we get 3 halves X plus,
and then 3 halves times 4 will just give us 6.
For this example, we want to find the equation of the line that contains the point 5 comma 1
and is parallel to the line X equals negative 6.
So this problem is going to be actually quite different
from the other examples that we were working on.
So we know that our line that we need
to find must be parallel to the line X equals negative 6.
So whenever you have an equation of the form X equals some number,
that's actually going to be a vertical line.
So let's go ahead and graph the line X equals negative 6.
So graphing the line X equals 6 negative, I know I must cross the X axis at negative 6.
So we get this vertical line.
So since my line needs to be parallel to this line,
I know that my line must also be a vertical line.
And it also has to go through the point 5 comma 1.
So plotting the point, 5 comma 1, and again knowing that it must be a vertical line,
let's draw the vertical line crossing the point 5 comma 1.
We get this line.
So then, the equation of this line, since it's crossing the X axis at 5,
the equation of this line will be X equals 5.
Pause the video and try these problems.