A laser is fired at a flat mirror.

The laser beam starts at the point (0,5) following the equation y = 5 - x/3 heading down to the right.
The mirror follows the equation y=x/2.

Find the equation for the path the beam takes after it hits the mirror.

OK, OK, these all work, but you seem to be making it way harder than it has to be. It's already been well established that the lines intersect at (6,3). Now, we just need to find the angle of intersection, which can be found by applying the law of cosines to the triangle formed by the two lines and the y-axis. The vertical side equals 5, and the other two are ¡î[(6-0)©÷+(5-3)©÷] and ¡î[(6-0)©÷+(3-0)©÷], which come out to be ¡î40 and ¡î45, respectively. We get:

25=40+45-2*¡î(45*40)*cos¥è

cos¥è turns out to equal ¡î2/2, so ¥è=45¨¬. This means that the two rays are perpendicular and their slopes are negative reciprocals. Since the first had a slope of -1/3, the second has a slope of 3.

The line becomes (y-3)=3(x-6), or, simplified into pussy point-slope formula, y=3x-15.