Answer:
![y=\frac{7}{4} x](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B7%7D%7B4%7D%20x)
Step-by-step explanation:
we have the original line equation
![7y + 4x = 3](https://tex.z-dn.net/?f=7y%20%2B%204x%20%3D%203)
clearing for y:
![7y=-4x+3\\y=\frac{-4}{7}x+ \frac{3}{7}](https://tex.z-dn.net/?f=7y%3D-4x%2B3%5C%5Cy%3D%5Cfrac%7B-4%7D%7B7%7Dx%2B%20%5Cfrac%7B3%7D%7B7%7D)
Now we have an equation of the form slope- intercept:
![y=mx+b](https://tex.z-dn.net/?f=y%3Dmx%2Bb)
where m is the slope and b is the y-intercept.
thus, the slope of the original line is:
![m=\frac{-4}{7}](https://tex.z-dn.net/?f=m%3D%5Cfrac%7B-4%7D%7B7%7D)
Now to find the new line, since it has to be perpendicular their slopes must satisfy the following:
![m*m_{1}=-1](https://tex.z-dn.net/?f=m%2Am_%7B1%7D%3D-1)
where m is the slope of the original line, and m1 is the slope of the new line:
![\frac{-4}{7}*m_{1}=-1\\ m_{1}=\frac{-1*7}{-4}\\ m_{1}=\frac{7}{4}](https://tex.z-dn.net/?f=%5Cfrac%7B-4%7D%7B7%7D%2Am_%7B1%7D%3D-1%5C%5C%20m_%7B1%7D%3D%5Cfrac%7B-1%2A7%7D%7B-4%7D%5C%5C%20m_%7B1%7D%3D%5Cfrac%7B7%7D%7B4%7D)
this is the slope of the new perpendicular line that passes trough the point (-4,-7), so now we use the point slope equation to find the equation of said line:
![y-y_{1}=m_{1}(x-x_{1})](https://tex.z-dn.net/?f=y-y_%7B1%7D%3Dm_%7B1%7D%28x-x_%7B1%7D%29)
where we know
, and from the point (-4,-7) ![x_{1}=-4, y_{1}=-7](https://tex.z-dn.net/?f=x_%7B1%7D%3D-4%2C%20y_%7B1%7D%3D-7)
so we have:
![y-(-7)=\frac{7}{4} (x-(-4))\\y+7=\frac{7}{4} (x+4)](https://tex.z-dn.net/?f=y-%28-7%29%3D%5Cfrac%7B7%7D%7B4%7D%20%28x-%28-4%29%29%5C%5Cy%2B7%3D%5Cfrac%7B7%7D%7B4%7D%20%28x%2B4%29)
and we clear for y to leave the equation in the slope intercept form:
![y=\frac{7}{4} (x+4)-7\\y=\frac{7}{4} x+\frac{7}{4}*4 -7\\\\y=\frac{7}{4} x+7-7\\y=\frac{7}{4} x](https://tex.z-dn.net/?f=y%3D%5Cfrac%7B7%7D%7B4%7D%20%28x%2B4%29-7%5C%5Cy%3D%5Cfrac%7B7%7D%7B4%7D%20x%2B%5Cfrac%7B7%7D%7B4%7D%2A4%20-7%5C%5C%5C%5Cy%3D%5Cfrac%7B7%7D%7B4%7D%20x%2B7-7%5C%5Cy%3D%5Cfrac%7B7%7D%7B4%7D%20x)