Hello there again. I am Mr. Tarrou. I want to talk to you again about finding equations of lines.
Unlike the other day, instead of using one point and and a slope or two points,
this time we are going to be given a point and an equation of another line...
or at least some information about another line.
So lets first make sure that we completely understand
the idea of parallel lines or slope and
perpendicular slopes.
Here we have a diagram of a white line that crosses through the origin and has a slope of one over three.
Here I have a parallel blue line.
Why are these two lines parallel besides the fact that they look very similar?
On this blue line to get from this point to this point you go up one and over three.
So this blue line has a slope of one third.
This white line has the same slope.
If I count out this slope, to get from this point to this point I have to go up one and over three.
So we have now shown that both of these lines have the exact same slope.
Now they do have different y intercepts.
So since we are talking about equations of lines,
Lets write over here.
This white line has a slope of one over three. Again up one and over three.
When you are writing in slope intercept form, it is y equals m...the slope... one third x plus B.
Now B is your y intercept and in this case we are crossing through the origin so our y intercept is zero.
So, our equation is y equals one third x.
Now the blue line that is parallel we can see that it crosses the y axis, that is the y axis that goes straight up and down,
at one, two, three.
So our y value or y intercept is at three.
This blue line's equation is y equals one third x , same slope because they are parallel...and we counted it out, our y intercept of 3 so we have
so we have y equals one third x plus three.
Now lets see what this looks like when we are just given information and not a graph.
We are going to find a line that is parallel to
3x plus 2y equals 10.
and goes through (4,6)
The other day I talked about the slope intercept form of a line y equals mx plus b,
the General Form or Standard Form which is Ax+By+C=0,
and then you have what we are going to be using which is point slope form for the information given.
Point-Slope form. Point Slope form tells you what you need to write the equation of a line.
You need a point which we have.
And you need a slope.
Unlike the other day, I am not going to just give you the slope.
We need to use this equation to come up with the slope. That is why the equation of this line is given to us.
With this we are going to find the slope we need to fill out the rest of this information.
This is in almost General Form, most books would call this Standard Form,
I cannot just look at this and tell what the slope is because y is not by itself.
So I need to work this out. I need to get the 3x over to the right hand side.
I need to get the 2 away from the y. When I have y equals mx plus b, then I will be able to look at the equation and see the slope.
I am going to move the 3x over to the other side with subtraction.
That bothers me, it looks like a division sign so I am going to get rid of this.
And we have 2y equals negative 3x plus 10.
I want to get the y by itself so I am going to divide everything by two.
And we get negative three halves x plus five.
The only purpose when a book gives you a point and another line...the only reason that other line is given to you is so that you can figure out the slope.
Well I have the slope now. The slope is negative three over two.
If I want a parallel line, I will use this slope the way it is.
If I want a perpendicular line, perpendicular lines have opposite reciprocal slopes,
I would have to flip this and change the sign.
But I want a parallel line!
The point I have and the slope I am going to use is right there. My slope is negative three over two.
Now I just said they give you an equation of another line so you can find a slope.
Well I just found it. You know what?
That means I do not need to look at the equation of the line given to me any more.
Any you really should erase that or scribble it out because a lot of students will be distracted by the numbers in that equation
even after they found the slope that was in there.
So make sure you don't look at it again.
We have a point and a slope.
If we have a point and a slope, we are going to plug into the point-slope form
which is y minus y sub 1 equals m times x minus x sub 1.
Don't forget that this is where our numbers will go.
x sub 1, y sub 1, and m
So we have y minus the y from the point equals the slope which is negative 3 over 2 times x minus the x from the point which is 4.
This is in point slope form. A lot of books will not be happy or your teacher will be happy if you leave it in this form.
So I am going to distribute this through and put into slope intercept form which I will ask my students to do.
negative 3 halves times one is negative 3 halves x
negative time negative is positive and three times four...
four over one, this is a fraction, so 3 times 4 is twelve and 12 divided by 2 is 6...so six.
I need to take this negative six, any time you move a number across the equal sign you need to do the opposite, so I am going to add 6 to both sides.
We get y equals negative 3/2 times x plus 12 and I have the equation of my parallel line:D
Alright. Lets try another one.
What if I want a line that is perpendicular to the given line?
Perpendicular to another line...so lets say I want a line
a line that is perpendicular to y equals one fourth x minus 4 and goes through the point of (2,1)
Now this equation that I have been given is already in slope-intercept form.
y equals mx plus b...y=mx+b
All you need to write the equation of a line is a point and we have that, and a slope.
My point is (2,1) and my slope, my slope here is equal to one fourth.
But I want a perpendicular slope and perpendicular slopes are opposite reciprocals.
The perpendicular slope to one fourth is negative four.
Let me take just a second and explain why perpendicular slopes are opposite reciprocals.
If I go up one and over four from the origin.
Just an example, I am not trying to graph this line.
Here is a line that has the slope of one fourth.
Let me take a book here and use it to make a drawing that is perfectly, as perfect as I can, perpendicular.
Now lets count this off.
What is the slope of this line? How do we get to this point to this point?
Well now I have to go down 1,2,3, 4 over 1.
When you go down or up that is a form of rise.
Over one, the our run is 1.
Here we have a quick, I should have put the chalk grid down, a quick visual sketch of what perpendicular means (looks like).
We can see why they are opposite reciprocals in this example.
At any rate, we have a point and a slope, and once again we have y minus 1 equals m times x minus x sub 1 which is x minus 2.
There is our perpendicular line.
Lets distribute this negative four through the parenthesis.
I am just taking the step again to put this into slope intercept form.
Negative times negative is positive.
Add both sides by one.
We get y is equal to negative 4x plus 9.
Perpendicular line.
Let me do one that is a little bit trickier. I have a few more minutes.
I want to find an equation of a line
that is through (-1,5) and is perpendicular to the line x equals four.
This line is a very special case. x equals a number is a vertical line. These lines have an undefined slope.
They also do not have a y intercept.
So if you do not understand or have memorized the properties of vertical lines, this is going to be an impossible question.
I also asked for the equation of a perpendicular line.
What is perpendicular to x = 4?
This is the line x=4.
Perpendicular to a vertical line is a horizontal line.
We are going to draw some kind of horizontal line. I almost drew a line there.
I want my horizontal line to go through negative one five. (-1,5)
(-1,5) Lets do this in blue.
Our point is right here, back one and up five.
So I want a line perpendicular to a vertical line that goes through that point.
If I draw a horizontal line through that point...oh...that is perpendicular and that is horizontal.
Horizontal lines have a slope of zero. The equation which is what the problem is asking for is... the equations of horizontal lines is y equals a number.
What y value does this number contain?
The x is negative one and the y is 5.
This again is a special case. Horizontal and vertical lines, if you do not have those memorized you will have trouble with those questions.
I am going to keep going and do one more example. Hopefully I do not run out of time.
I want a line through (0,4) parallel to a line with an x intercept of seven and a y intercept of negative 5.
Again a sort of weird one. This is not your typical problem from your textbook since we are only given x and y intercepts.
You can't look at this and immediately tell what the slope is.
We have a point but we need a slope.
I need to get the slope from these x and y intercepts.
You might be able to figure out the slope off the top of your head (mental math),
I like to, with geometry questions especially when the are difficult, draw a picture.
I have a line with an x intercept of 7. 1,2,3,4,5,6,7
I have a y intercept of negative five. 1,2,3,4, negative 5
Now...ok...so... Now that I have this drawn I have a line whose slope is up five and over seven.
Now I have my point and my PARALLEL slope of five over seven.
Now I have done a couple of examples completely through, and I am running out of time so I am going to short cut this a little bit.
Do you see how this point has an x value of zero?
Any point with an x value of zero means you did not move off the origin, we just moved up four places.
Well if you have just moved up four places, you are still on the y axis!
If you are on the y axis, that means that this is a y intercept.
And if this is the y intercept already, and we just drew a picture to find the slope, ... slope intercept?
I think that is y equals mx plus b. I can write this answer in slope intercept form with out any extra work at all. That is a big ol fat BAM!!!
Have a great day!