So in this lesson
We will be and trying to understand the straight lines. How state lines behave
What are their properties and how we can understand straight lines in a simple and efficient manner?
so first of all
I would like to mention that this lecture is for your reference
Whenever you have any problems in understanding the state lines or solving problems of state lines
You can come back and you can have a look at this lecture. So what is a line whenever we connect two points?
We make a line
These points can be very close to each other. These points can be very far from each other and
This line is an infinite line. So this point can be very close
Very tiny distance between them even then this line will be infinite
Because when we say line so line means an infinite straight lines
Which goes in the direction in which we define it?
So for example this line I define using two points P 1 is minus 1 4
minus 1 on the x axis
Y along the Y for all of the y axis for this the other point I took for this line was P 2 which is
4 minus 6 4 units
Along the x axis and 6 units in the negative y direction negative 6
So I just joined these two points and then this line is going infinite distance similarly
I joined these two points and this line is going
Infinite distance. So no if you look at the slope of this line
So how you get the slope of any line or how you draw a line if the slope is given
So this is the first thing we try to understand so
To in order to get the slope
What you can do is you start from any point so you can go right you can go left if you go left you
Go parallel to x axis. This is axis. And this is parallel to x axis you go towards left and there you go
downward and you reached at the line again and you measure the distance in the extraction and in
Direction then you take the ratio of change in Y
divided by X and you get the slope in the previous lecture we
Already derived the equation of this straight line
So this is the equation of this line y equals x plus 2 if you compare this equation with the equation of the line
MX plus C. You will find that M is 1 here
M is 1 and you can see that this is one unit in the Y Direction divided by 1 units in the end
Extraction you will get 1 so this is the equation of the straight line
Now this is the point where this line intersects
Y axis, so this is the
Distance from 0 to this point which is 2 units 1 & 2 2 unit is the intercept y intercept of this line
Which is highlighted here
Y intercept of this line is 2 units. This is one intercept
Now if you look at another line this one now this line what what is the slope of this line?
This line has an equation y is equal to minus 2x plus 2, for example, you don't know this equation and you are asked to
Derive, the equation of this line you are given only two points
P1 and p2. So how will you reach this equation?
just find the slope of this line and
The point where the line intersects the y-axis
So the line intersects the Y X is at 2 so you will proceed like this you have p1
For example, you were given only p1
minus 1/4 and
You are given p2
Somebody asked you that these are the two points
Which makes the line you connect these two points to make a line or they say that these two points lie on the line
Find the slope of the line. So what will be the slope slope will be equal to difference in Y coordinates
so you can take the difference in both ways like
Delta Y
Divided by Delta X. So what you do if you are taking Y - this is this is for example
This is point one
so we say this is X 1 and this is y one and
point two we say this is X 2 and this is y 2
No, remember this Delta Y you can take like this. Y2
Minus y1
Divided by x2 minus x1 and
If you take the minus common from this
- common then this will become minus. Y2
Because the sign will change and this will become plus. Y1
And you take the minus comment from the denominator. I
Just wanted to start to show you something therefore I am doing this
Otherwise you can straight away calculate, but I want to show something
If you take minus common, you will be left with minus X 2 plus
X 1 now
If you simplify this
- and this -
both - men divided it will become plus just like
Multiplication when minus x minus is plus minus - added by minus is also plus
So then you will be left by
No, I you can change the position. But I first write the same way - y 2 plus y 1
Divided by minus X 2 plus X 1 now
you can switch the position because two terms are adding a plus B or B plus a is same so we can say
Y 1 minus y 2 and
X 1 minus X 2
Now look at this equation and
This equation. This is the source of error because student do here a lot of errors if you take the
Y 2 minus y 1 then you must take X 2 minus X 1
You take first point as P 1
And it's coordinate as x 1 y 1 and take the second point P 2 and it's coordinates X 2 y
No for delta-y, either you take Y 2 minus 1 and X 2 minus X 1 how you take?
Y 1 minus y 2 but you have to take X 1 minus X 2 here. Otherwise the sign will change
If you take once y 2 minus y 1 and divide it by X 1 minus X 2
The sign will the sign of the slope will change
So instead of line going in this direction the - slope means it will be going in this direction
Yes, I am going to explain next but so just keep this in mind
So no we decided this that we will use this formula
Y2 minus y1 so then this is our y2 so minus 6
Minus minus y bar is 4
Divided by x2 which is 4 minus this minus 1
So minus 6 and minus 4 is minus 10
For this will become plus 1 4 plus 1 is 5
No 10 divided by 5 is 2 + - plus is minus
So minus 2 is the slope so now you can see here that minus 2 is the M
So know what I can have either a positive slope or a line can have a negative slope
So what type of line will have a positive slope so if you look at this if the line is going like this?
whether it's
Whatever may be the y-intercept
If even this one are the other line could be this one
Or it could change like this one in all of these situations when the line is going
Towards the right and left like this
Upwards end on the leftward direction you can see the slope will always be positive
because from any point
Let's say for this line if I go
Negative x-direction I am going negative x direction and negative Y direction so negative Y
Divided by negative X will make this positive
so if I go positive
Then I have to go positive in order to reach the line. So it means positive
Divided by positive will make this slope positive
Now the lines which are going in this direction
What will happen like this one here. If I start from this point in order to find the slope and go this way?
So this is negative x so negative
Then I have to go positive in the Y
So positive Y
Because slope is the ratio of the change. So positive Y and negative X
The slope will be negative
and even if to go on the other direction, for example here you can do like this if for example you go
ye positive like this and
Then you have to go X negative
So same so this line even if you go in this coordinate what you will be doing you go first X negative
Then Y positive
so in any case if the line is going in this direction even a little bit of
angle like this
Even a little bit of angle like this the slope will be negative
and if the line is going towards like this upward direction, then the slope will be
Positive so these are few things. What does equation of line means when we say y equals x plus 2
We know that low power intercept what other information we can get from here
so what does it mean is that for example, I have X and
I have few values of X 1
2 3 4 and that's the only 4 values I
Try for well, so what will be the values of Y?
so if I put X 1
when x equals 1 y equals
1 plus 2 3 so it means this point 1 & 3
Will be on the line and if you look here you will see 1 in the X direction and 1 2 & 3
This might actually lies on the line
So for every value of x you can find the value of y coordinate on this line
So this is what they because this is sometimes called the characteristic equation of the line
character extreme characteristic
Equation of the line because if you know the value of x you can find the value of y
Anywhere on the line x axis is also a line. This is a line. So what will be the slope of x axis so
X-axis has a zero slope
X-axis has a zero slope because there is no change in Y
Slope is defined as because this line if I take any true point on this line, for example, I take this point and this point
So what is Delta Y between this point?
Delta Y is zero between these two point and what is Delta X?
Delta X is 1 so what will be the slope of x-axis?
Delta Y divided by Delta X which is 0 divided by 1 which is 0
Now come to y-axis. What will be the slope of Y X is y axis is also a line y axis is a line
Isn't it YX is a line so what is the slope of y axis so what is slope slope is
Delta Y
between any two points divided by Delta X between any two points change in Y coordinate between any two points
So let's say I take this point and I take this point
So what is the y coordinate of this point 6
So I say 6 minus what is the y coordinate of this point 3
Okay, I got the change in Y. What is the x coordinate of this point?
0 because this is on my Y axis
So
Zero - what is the coordinate of this point? What is the x-coordinate of this point is zero?
So it means the slope will be
3 divided by zero and
What is if you divide a number with zero what you get?
infinite
So the slope of y axis is infinate
The slope of the y-axis infinate
So what is the characteristic equation of y axis what is the equation which describes?
Y axis so we say that characteristic equation of y axis is x equals 0
On y axis x equals 0 this is the equation of the y axis you can see that on all y axis x is 0
similarly, you can see the equation of the x axis is
y equals 0
This is the equation of the x axis or all x axis y is 0 so this is characteristic equation of these X's
For example take this line. There is a line which is parallel to x axis
This line so what will be the equation of this line
On all this line
Because this is a this is a line parallel to x axis
So you would think okay the x axis has an equation y equal to 0
So this line should also have the same type of equation. Yes it is because it has y equals 1
All this line y is equal to 1
So equation of the line mean the equation must be true for every point of the line
So only this y equals 1 can be true for every point of this line
No other equation can be true for this line. Similarly. The line is like this if the line is parallel to Y axis
If a line is parallel to Y axis
So it must have an equation something like this x equals
One two three, but minus three
X equals minus three is a line
Which is like this, which is parallel to X
so if you see a line equation, if you if the normal equation of the line is this
But if you see the equation of the line is given as X minus 3
So
You can also say okay. This is an equation of the line, but this line is
Parallel to y axis because there is no y in this
So
But these are special special
Equation of the lines you can say these are the special
normally whenever you talk about
Straight line you say straight line has an equation of y equals MX plus C. So this is the normal
Equation of the straight line. I am just trying to enhance your knowledge a little bit so that you can better understand straight lines
That straight lines can be of any type. So if a straight line is in this direction
Even so the slope will be because there can be only two directions one. Is this one the other is this one?
Otherwise if the line is straight then it's parallel to x axis and
Otherwise if the line is vertical it is parallel to Y axis so then it will have equation something like this
So this is what you should understand that straight lines or straight lines behave
No, I will show you what are perpendicular lines and what are parallel lines