Given Information

#y - 2x = 3# need to find a perpendicular line to this line that passes through the point #(2, 5)#

For starters, solve for #y# in the first equation.

#y - 2x = 3#

#y = 2x + 3#

Now that we have the equation that is easy to read, to make a line perpendicular, the slope is ALWAYS the reciprocal. So the slope in this equation is #2x#. The reciprocal, which is the absolute opposite in sign and numbers (flipping up side down as some call it), would be #- 1/2x#. So if you were to graph the lines

#y = 2x + 3# and #y = -1/2x + 3# they would be perpendicular. However, we are not done yet, since we need to get it through the point #(2, 5)#.

Doing this, you need to use the equation #y - h = m(x - k)#. Your teacher may have used this equation, but with different variables.

#y and x# will be staying the same so the only variables you need to worry about would be #h, m, and k#.

#h# will the the #y# coordinate of the point you want it to pass through, in this case, that would be #5#, since in #(2,5)#, #5# is the #y# coordinate.

#m# will be the slope, we already calculated that as it's the reciprocal, so it'll be #-1/2#.

#k# is the #x# coordinate of the point. In #(2, 5)#, the #x# coordinate is #2#.

So there you have it! Just plug in the known information in the equation.

#y - h = m(x - k)# Equation

#y - (5) = -1/2(x - 2)#

#y - (5) = -1/2x + 1# || Distributed the #-1/2# to the #x# and #-2# term.

#y = -1/2x + 6# Added the #5# from the left side.