Find point on the line

The shortest distance from a point to a line is the perpendicular distance from that point to that line.

 

Step 1:  Find the slope of the line  -6x + 5y - 1  =  0.  Call this slope m₁.

Step 2:  Find the negative reciprocal of that slope; call this slope m₂.

                 This is the slope of the line that is perpendicular to the original line.

Step 3:  Use the point-slope form ( m₂ and the point (-3,1) ) to find the equation of the line through

                 the point (-3,1) that is perpendicular to the original line.

Step 4:  Using the original equation and the equation that you found in step 3, find the point of 

                 intersection of these two lines. This is the answer.