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.

The statement "Taking that further, the slope of the laser beam is -1/3, so after reflection the slope is 3." assumes that the reflected beam is going to be perpendicular to the incident beam. That would be the case if the beam is incident at 45 degrees to the mirror (which it in fact is) but I don't think that was established yet.

My derivation is:

The mirror is sloped upward arctan(1/2) and the bouncing photons downward at arctan(1/3). The angles these make with the y-axis, on the inside of the triangle formed by the y-axis, the mirror and the light beam, are 90 degrees minus each of these, thus adding up to 180 degrees minus the sum of these.  Therefore the sum of the two arctans is the third angle, between the light beam and the mirror. Arctan(1/2) is 26.56505117707799 degrees and arctan(1/3) is 18.43494882292201 degrees.  These add to 45 degrees.

So the reflected beam is traveling upward at arctan(1/2) + 45 degrees, or 71.56505117707799 degrees. This angle's tangent is 3, which is then the slope of its line.  It goes through point (6,3) (the intersection of the light beam and the mirror), and so the equation for the path of the reflected beam is y-3 = 3(x-6), or y = 3x - 15.