First, let's establish what we know. We know that the line we're trying to find is perpendicular to the line given by the equation y = -5x + 8 and it passes through the point P(3,4).
We also know that for a line to be perpendicular to another, the product of their slopes must equal -1. This is a fundamental property of perpendicular lines in a coordinate system.
So, given that we have the slope of the initial line (which is -5), we can calculate the slope of the line that's perpendicular to it by taking the negative reciprocal of the initial slope.
That is, slope_required = -1/slope_given = -1/(-5) = 1/5 = 0.2
Now, we know the slope of the line (m) we're looking for and we also have a point (x, y) that lies on our line. We can use these to compute the y-intercept (c) of our line by rearranging the formula for a line, y = mx + c.
We substitute m = 1/5, x = 3, and y = 4 into the equation to get 4 = 1/5 * 3 + c.
Solving this equation for c gives us c = 4 - 1/5 * 3 = 3.4
So, the equation of the line that is perpendicular to y = -5x + 8 and passes through the point P(3, 4) is y = 1/5 * x + 3.4.
Therefore, the correct choice is option c) y = 1/5 * x + 3.4.