How do you solve #a/(ax-1)+b/(bx-1) = a+b# ?

1 Answer
Jul 1, 2017

#x = 0" "# or #" "x = (a^2+4ab+b^2)/(a^2b+ab^2)#

Explanation:

Given:

#a/(ax-1)+b/(bx-1) = a+b#

Subtract #a/(ax-1)+b/(bx-1)# from both sides to get:

#0 = a+b-a/(ax-1)-b/(bx-1)#

Multiply through by #(ax-1)(bx-1)# to get:

#0 = (a+b)(ax-1)(bx-1)-a(bx-1)-b(ax-1)#

#color(white)(0) = (a+b)(abx^2-(a+b)x+1)-a(bx-1)-b(ax-1)#

#color(white)(0) = (a+b)(abx^2-(a+b)x)+color(red)(cancel(color(black)((a+b))))-2abx-color(red)(cancel(color(black)((a+b))))#

#color(white)(0) = (a+b)(abx^2-(a+b)x)-2abx#

#color(white)(0) = x((a+b)(abx-(a+b))-2ab)#

#color(white)(0) = x((a+b)abx-((a+b)^2+2ab))#

#color(white)(0) = x((a^2b+ab^2)x-(a^2+4ab+b^2))#

#color(white)(0) = (a+b)abx(x-(a^2+4ab+b^2)/(a^2b+ab^2))#

Hence:

#x = 0#

or:

#x = (a^2+4ab+b^2)/(a^2b+ab^2)#