We start with an equation in standard form and we need to find the equation of the line perpendicular that goes through (8,3). We need our equation to be in slope-intercept form (y = mx + b)
The first thing to recognize is that we need to use the point-slope equation because we need to go through a specific point. In order to use the point-slope equation, we need two things: (1) the point the line goes through and (2) the slope of the line. Now we have the point that the line goes through, but we don't have the slope of the new line.
If we want to describe a line perpendicular to our original line, the slope of our new line will be the negative inverse of the original slope. m2 = -1/(m1)
But what is the slope (m1) of our original line? If we put the original line into slope-intercept form, we get
y = -x/6 + 8 => m1 = -1/6
Taking the negative inverse of the original slope, we get m2 = 6
Now that we have the slope of our second line, we can use the point-slope form to construct our equation.
y - y1 = m2(x - x1) => y - 3 = 6(x - 8) => y - 3 = 6x - 48
Now we solve for y to get our equation into slope-intercept form.
y = 6x - 45