Perpendicular lines intersect to form a 90° angle. They form a sort of X-shape.
The slope of a line perpendicular to another line is the negative reciprocal, that is,
![m_{perp} = \frac{-1}{m}](https://tex.z-dn.net/?f=m_%7Bperp%7D%20%3D%20%20%5Cfrac%7B-1%7D%7Bm%7D%20)
.
For your equation, the slope is –7. That means the slope of its perpendicular line must be
![- \frac{-1}{7} = \frac{1}{7}](https://tex.z-dn.net/?f=-%20%5Cfrac%7B-1%7D%7B7%7D%20%20%3D%20%20%5Cfrac%7B1%7D%7B7%7D%20)
.
Now, we must also find such a line that passes through (7, 5). It is easiest to write this in point-slope form,
![y-y_1=m(x-x_1)](https://tex.z-dn.net/?f=y-y_1%3Dm%28x-x_1%29)
.
We get
![y-5= \frac{1}{7} (x-7)](https://tex.z-dn.net/?f=y-5%3D%20%5Cfrac%7B1%7D%7B7%7D%20%28x-7%29)
. Technically, we are done, but it is customary to convert this into slope-intercept form,
![y=mx+b](https://tex.z-dn.net/?f=y%3Dmx%2Bb)
.
![y-5= \frac{1}{7} (x-7)\\\\y= \frac{1}{7}x-1+5\\\\y= \frac{1}{7}x+4](https://tex.z-dn.net/?f=y-5%3D%20%5Cfrac%7B1%7D%7B7%7D%20%28x-7%29%5C%5C%5C%5Cy%3D%20%5Cfrac%7B1%7D%7B7%7Dx-1%2B5%5C%5C%5C%5Cy%3D%20%5Cfrac%7B1%7D%7B7%7Dx%2B4)
.
And that's our final answer. We can verify the solution graphically. See attached.
In general, our steps are:
<span><span>1) Find the slope of the given line if it is not given directly.
</span><span>2) Find the slope of the perpendicular line (the negative reciprocal).
</span>3) Identify the point it must pass through.
<span>4) Substitute the values into the point-slope form.
</span><span>5) Write the equation in slope-intercept form.</span></span>