Students reading in class IX and X can send mathematics , which they are not able to understand or grasp, send an email with the mathematics at
kaacconsultant@gmail.com
that math will appear in this blog which will help others also
Two liner equation, both represents straight line
a1 x + b1 y + c1 = 0
and a2 x + b2 y +c2 = 0
Both the equation represents two straight lines
General equation of an straight line Y= M X + C where M is the inclination to the vertical axis called gradient or slope and C is the point where the line intersects with vertical axis.
If we try to rearrange the equations in Y= MX + C form
a1x + b1y +c1 = 0
or , b1y = -a1x - c1
or y = -a1/b1 x - c1/b1
Now the equation took the shape of Y = MX +C
so M= -a1/b1
and C = -c1/b1
Suppose two linear equation are as follows
6x-2y +9 = 0 ------------ 1
3x - y +12 = 0 ------------ 2
equation 1 6x -2y +9 = 0
or, -2y = -6x -9
or y = -6/-2 x -9/-2
or y = 3 x +9/2
From equation 1 slope or gradient value is 3
For equation 2, 3x - y + 12 =0
or -y = -3x -12
or y = 3x +12
In this equation slope or gradient is also 3
So both the equations gradient is same so the lines are parallel.
When two lines are parallel , both the lines can not intersect at a common point. So the intersection point is unreal.
In the other example
Two equations are 2x +3y -2 = 0 ------------- 1
x - 2y - 8 = 0 ------------2
slopes or gradients are 2x +3y -2 = 0
or, 3y = -2x +2
or, y = -2/3 x + 2/3
so slop is -2/3
For second equation x -2y -8 = 0
or, -2y = -x +8
or y = 1/2 x - 8/2
so slop of the second line is 1/2
When both the lines slope that is angle of inclination or gradient is different the lines will intersect at a common point
Now if we try to solve the equation we will get the point of intersection that is at the point of intersection both the equation is satisfied
From first equation y = -2/3 x + 2/3
From second equation y= 1/2 x - 8/2
if the value of y is put in equation 1
1/2 x -8/2 = -2/3 x + 2/3
or, 1/2 x + 2/3 x = 2/3 + 4
or, x = 4
Putting x =4 ,
y = 1/2*4 - 4
or y = -2 So the point of intersection (4,-2 )
Now in other conditions , let two lines are coincidence that is one line is lying on other , then how come the equations are different, It happens , when one line length is bigger or shorter than the other line.
Two equations are 5x -15y = 8 ------------ 1
other equation is 3x - 9y = 24/5 -----------2
slope or gradient will be same and simultaneously intersection point with the vertical axis will also be the same
rearranging the lines in y= mx +c pattern
5x -15 y = 8
or, -15 y = -5x +8
or, y= 1/3 x - 8/15
For other equation 3x -9y = 24/5
or, -9y = -3x + 24/5
or, y = 1/3 x - 8/15
So for both the equations slope and intersection point at the vertical axis are same . So both the lines are coincidence.
Now another interesting aspect
If the equation x = a this is also an equation of a straight line which is parallel to vertical y axis
Similarly if the equation is y = b , this straight line is parallel to horizontal x axis. .
this both straight lines will met at the intersection point at (a,b)
kaacconsultant@gmail.com
that math will appear in this blog which will help others also
Two liner equation, both represents straight line
a1 x + b1 y + c1 = 0
and a2 x + b2 y +c2 = 0
Both the equation represents two straight lines
General equation of an straight line Y= M X + C where M is the inclination to the vertical axis called gradient or slope and C is the point where the line intersects with vertical axis.
If we try to rearrange the equations in Y= MX + C form
a1x + b1y +c1 = 0
or , b1y = -a1x - c1
or y = -a1/b1 x - c1/b1
Now the equation took the shape of Y = MX +C
so M= -a1/b1
and C = -c1/b1
Suppose two linear equation are as follows
6x-2y +9 = 0 ------------ 1
3x - y +12 = 0 ------------ 2
equation 1 6x -2y +9 = 0
or, -2y = -6x -9
or y = -6/-2 x -9/-2
or y = 3 x +9/2
From equation 1 slope or gradient value is 3
For equation 2, 3x - y + 12 =0
or -y = -3x -12
or y = 3x +12
In this equation slope or gradient is also 3
So both the equations gradient is same so the lines are parallel.
When two lines are parallel , both the lines can not intersect at a common point. So the intersection point is unreal.
In the other example
Two equations are 2x +3y -2 = 0 ------------- 1
x - 2y - 8 = 0 ------------2
slopes or gradients are 2x +3y -2 = 0
or, 3y = -2x +2
or, y = -2/3 x + 2/3
so slop is -2/3
For second equation x -2y -8 = 0
or, -2y = -x +8
or y = 1/2 x - 8/2
so slop of the second line is 1/2
When both the lines slope that is angle of inclination or gradient is different the lines will intersect at a common point
Now if we try to solve the equation we will get the point of intersection that is at the point of intersection both the equation is satisfied
From first equation y = -2/3 x + 2/3
From second equation y= 1/2 x - 8/2
if the value of y is put in equation 1
1/2 x -8/2 = -2/3 x + 2/3
or, 1/2 x + 2/3 x = 2/3 + 4
or, x = 4
Putting x =4 ,
y = 1/2*4 - 4
or y = -2 So the point of intersection (4,-2 )
Now in other conditions , let two lines are coincidence that is one line is lying on other , then how come the equations are different, It happens , when one line length is bigger or shorter than the other line.
Two equations are 5x -15y = 8 ------------ 1
other equation is 3x - 9y = 24/5 -----------2
slope or gradient will be same and simultaneously intersection point with the vertical axis will also be the same
rearranging the lines in y= mx +c pattern
5x -15 y = 8
or, -15 y = -5x +8
or, y= 1/3 x - 8/15
For other equation 3x -9y = 24/5
or, -9y = -3x + 24/5
or, y = 1/3 x - 8/15
So for both the equations slope and intersection point at the vertical axis are same . So both the lines are coincidence.
Now another interesting aspect
If the equation x = a this is also an equation of a straight line which is parallel to vertical y axis
Similarly if the equation is y = b , this straight line is parallel to horizontal x axis. .
this both straight lines will met at the intersection point at (a,b)
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