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, [tex]m_{perp} = \frac{-1}{m} [/tex].
For your equation, the slope is –7. That means the slope of its perpendicular line must be [tex]- \frac{-1}{7} = \frac{1}{7} [/tex].
Now, we must also find such a line that passes through (7, 5). It is easiest to write this in point-slope form, [tex]y-y_1=m(x-x_1)[/tex].
We get [tex]y-5= \frac{1}{7} (x-7)[/tex]. Technically, we are done, but it is customary to convert this into slope-intercept form, [tex]y=mx+b[/tex].
[tex]y-5= \frac{1}{7} (x-7)\\\\y= \frac{1}{7}x-1+5\\\\y= \frac{1}{7}x+4[/tex].
And that's our final answer. We can verify the solution graphically. See attached.
In general, our steps are:
1) Find the slope of the given line if it is not given directly.
2) Find the slope of the perpendicular line (the negative reciprocal).
3) Identify the point it must pass through.
4) Substitute the values into the point-slope form.
5) Write the equation in slope-intercept form.
