>>13575013if you multiply by the denominator to the left on both sides, you obtain an equivalent equation except at the zero of the original denominator, which would not be defined and is added to the new expression
so if you have say A = B where A and B are expressions, and you multiply it by C on both sides, then the new equation A*C = B*C would generally have the same solutions as A = B except if by multiplying by C you altered the range
in this case the solutions to:
(x-1) / (x^2 -2x +1) = 3
and:
(x-1) = 3*(x^2 -2x +1)
=> 3x^2-7x+4=0
are the same except in the first one there are points that are undefined if the denominator has zeroes
so by solving the second one you are solving the first one except for those points
basically what you do when you multiply on both sides is to obtain two functions that intersect in the same points even if these functions are different, with the exception that you may be adding points to the range and these points may happen to be solutions
this doesn't happen when you add on both sides
when you divide on both sides you may be removing points from the range
(
exercise: what happens if you square both sides of an equation? think of solutions as intersections of the graphs of the functions on each side
spoiler example:
x = 1
vs
x^2 = 1^2
)
