A proportion is a statement that two ratios are equal to each other.
For example #3/6=5/10# (We sometimes read this "3 is to 6 as 5 is to 10".)
There are #4# 'numbers' (really number places) involved. If one or more of those 'numbers' is a polynomial, then the proportion becomes a rational equation.
For example: #(x-2)/2=7/(x+3)# ("x-2 is to 2 as 7 is to x+3").
Typically, once they show up, we want to solve them. (Find the values of #x# that make them true.)
In the example we would "cross multiply" or multiply both sides by the common denominator (either description applies) to get:
#(x-2)(x+3)=2*7#. Which is true exactly when
#x^2+x-6=14# Which in turn, is equivalent to
#x^2+x-20=0# (Subtract 14 on both sides of the equation.)
Solve by factoring #(x+5)(x-4)=0#
so we need #x+5=0# or #x-4=0# the first requires
#x=-5# and the second #x=4#.
Notice that we can check our answer:
#(-5-2)/2=-7/2# and #7/(-5+3)=7/-2=-7/2#. So the ratios on both sides are equal and the statement is true.
