Given 2 equations, why can I add and subtract them to obtain the 2 equations of the bisectors? Why this works

Given two equations, why can I add and subtract them to get the equations of the bisectors?





Example:


4x-3y=5 and 3x+4y=10; find the equations of the two bisectors of the angles formed by the two lines by adding and subtracting the two equations. Why this works? Is there a proof?Given 2 equations, why can I add and subtract them to obtain the 2 equations of the bisectors? Why this works
It's not true that adding or subtracting such linear equations will get you any of the bisectors. For example, adding





y + x = 0


y + 2x = 0





Gets you 2y + 3x = 0, with a slope of -1.5, which may be the average of slopes -1 and -2, but that is not an angle bisector. Your example happens to work only because of symmety reasons, the slope of the first is -4/3, the second is -3/4, so that the bisector would have the slope -1, which is what happens when you add the equations. Subtracting them gets a line of slope 1, which is perpendicular.





Addendum: About the only time this ';trick'; can work is if you have equations of the form ax + by = 0 and bx + ay = 0.