<u>Answer:
</u>
Equation of a line that is perpendicular to y = 3x-4 and that passes through the point (2 -3) is ![y=-\frac{1}{3} x-\frac{7}{3}](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B1%7D%7B3%7D%20x-%5Cfrac%7B7%7D%7B3%7D)
<u>Solution:
</u>
The slope - intercept form equation of line is given as
y = mx+c ----- (1)
Where m is the slope of the line. The coefficient of “x” is the value of slope of the line.
Given that
3x -y = 4
Converting above equation in slope intercept form,
y = 3x - 4 ---- (2)
On comparing equation (1) and (2) we get slope of equation (2) is m=3
Consider equation of the line which is perpendicular to equation (2) is
--- eqn 3
If two lines having slope m1 and m1 are perpendicular then relation between their slope is ![m_{1} m_{2}=-1](https://tex.z-dn.net/?f=m_%7B1%7D%20m_%7B2%7D%3D-1)
That is if slope of the line (2) is 3 then slope of equation (3) is ![m_{1}=-\frac{1}{3}](https://tex.z-dn.net/?f=m_%7B1%7D%3D-%5Cfrac%7B1%7D%7B3%7D)
On substituting value of m1 in equation (3), we get
--- eqn 4
Given that equation (4) passes through (2, -3) that is x = 2 and y = -3. So on substituting value of x and y in equation (4),
![-3=-\frac{2}{3}+b](https://tex.z-dn.net/?f=-3%3D-%5Cfrac%7B2%7D%7B3%7D%2Bb)
On simplifying above equation,
![\begin{aligned} b &=\frac{2}{3}-3 \\ b &=\frac{2-9}{3} \\ b &=-\frac{7}{3} \end{aligned}](https://tex.z-dn.net/?f=%5Cbegin%7Baligned%7D%20b%20%26%3D%5Cfrac%7B2%7D%7B3%7D-3%20%5C%5C%20b%20%26%3D%5Cfrac%7B2-9%7D%7B3%7D%20%5C%5C%20b%20%26%3D-%5Cfrac%7B7%7D%7B3%7D%20%5Cend%7Baligned%7D)
On substituting value of b in equation (4),
![y=-\frac{1}{3} x-\frac{7}{3}](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B1%7D%7B3%7D%20x-%5Cfrac%7B7%7D%7B3%7D)
Hence equation of a line that is perpendicular to y=3x-4 and that passes through the point (2 -3) is ![y=-\frac{1}{3} x-\frac{7}{3}](https://tex.z-dn.net/?f=y%3D-%5Cfrac%7B1%7D%7B3%7D%20x-%5Cfrac%7B7%7D%7B3%7D)