Equation of a line given ![7x-2y = 1](https://tex.z-dn.net/?f=%207x-2y%20%3D%201%20)
We have to find an equation of a line which is perpendicular to the given line.
If the general equation of a line is
, m is the slope of the line there. And the slope of the perpendicular line will be the negative reciprocal of m which is
.
So first here we have to make the given equation as ![y=mx+c](https://tex.z-dn.net/?f=%20y%3Dmx%2Bc%20)
![7x-2y = 1](https://tex.z-dn.net/?f=%207x-2y%20%3D%201%20)
First we have to move 7x to the right side by subtracting it from both sides.
![-7x+7x-2y =-7x+1](https://tex.z-dn.net/?f=%20-7x%2B7x-2y%20%3D-7x%2B1%20)
![-2y = -7x+1](https://tex.z-dn.net/?f=%20-2y%20%3D%20-7x%2B1%20)
Now to get y we have to move -2, by dividing it on both sides.
![\frac{(-2y)}{(-2)} = \frac{(-7x+1)}{(-2)}](https://tex.z-dn.net/?f=%20%5Cfrac%7B%28-2y%29%7D%7B%28-2%29%7D%20%3D%20%5Cfrac%7B%28-7x%2B1%29%7D%7B%28-2%29%7D%20%20)
![y = \frac{(-7x)}{(-2)} + \frac{1}{(-2)}](https://tex.z-dn.net/?f=%20y%20%3D%20%5Cfrac%7B%28-7x%29%7D%7B%28-2%29%7D%20%2B%20%5Cfrac%7B1%7D%7B%28-2%29%7D%20%20)
![y= (\frac{7}{2} )x - \frac{1}{2}](https://tex.z-dn.net/?f=%20y%3D%20%28%5Cfrac%7B7%7D%7B2%7D%20%29x%20-%20%5Cfrac%7B1%7D%7B2%7D%20%20)
So here the slope ![m= \frac{7}{2}](https://tex.z-dn.net/?f=%20m%3D%20%5Cfrac%7B7%7D%7B2%7D%20%20)
Now the slope for the perpendicular equation is
![-\frac{1}{m}= -\frac{1}{(7/2)}](https://tex.z-dn.net/?f=%20-%5Cfrac%7B1%7D%7Bm%7D%3D%20-%5Cfrac%7B1%7D%7B%287%2F2%29%7D%20%20%20)
![-\frac{1}{m} = -\frac{2}{7}](https://tex.z-dn.net/?f=%20-%5Cfrac%7B1%7D%7Bm%7D%20%3D%20-%5Cfrac%7B2%7D%7B7%7D%20%20)
So slope of the perpendicular line is ![-\frac{2}{7}](https://tex.z-dn.net/?f=%20-%5Cfrac%7B2%7D%7B7%7D%20%20)
We can write the perpendicular equation as
![y=(-\frac{2}{7} )x+c](https://tex.z-dn.net/?f=%20y%3D%28-%5Cfrac%7B2%7D%7B7%7D%20%29x%2Bc%20)
Now this equation is passing through the point (-4,5)
We have to plug in x = -4 and y = 5 in the line to get c.
![5= (-\frac{2}{7})(-4) + c](https://tex.z-dn.net/?f=%205%3D%20%28-%5Cfrac%7B2%7D%7B7%7D%29%28-4%29%20%2B%20c%20%20)
![5= \frac{8}{7} +c](https://tex.z-dn.net/?f=%205%3D%20%5Cfrac%7B8%7D%7B7%7D%20%2Bc%20)
![5-\frac{8}{7} = c](https://tex.z-dn.net/?f=%205-%5Cfrac%7B8%7D%7B7%7D%20%3D%20c%20)
![\frac{35}{7} -\frac{8}{7} =c](https://tex.z-dn.net/?f=%20%5Cfrac%7B35%7D%7B7%7D%20-%5Cfrac%7B8%7D%7B7%7D%20%3Dc%20)
![\frac{27}{7} =c](https://tex.z-dn.net/?f=%20%5Cfrac%7B27%7D%7B7%7D%20%3Dc%20)
So we have got the value of c. Now we can write the perpendicular equation as,
![y= (-\frac{2}{7})x+\frac{27}{7}](https://tex.z-dn.net/?f=%20y%3D%20%28-%5Cfrac%7B2%7D%7B7%7D%29x%2B%5Cfrac%7B27%7D%7B7%7D%20%20%20)
![y = \frac{(-2x+27)}{7}](https://tex.z-dn.net/?f=%20y%20%3D%20%5Cfrac%7B%28-2x%2B27%29%7D%7B7%7D%20%20)
![7y = -2x+7](https://tex.z-dn.net/?f=%207y%20%3D%20-2x%2B7%20)
![7y+2x = -2x+2x+7](https://tex.z-dn.net/?f=%207y%2B2x%20%3D%20-2x%2B2x%2B7%20)
![2x+7y = 7](https://tex.z-dn.net/?f=%202x%2B7y%20%3D%207%20)
So we have got the required perpendicular line.
The equation of the perpendicular line is ![2x+7y = 7](https://tex.z-dn.net/?f=%202x%2B7y%20%3D%207%20)