If a(x+b)=bx-c, what is x?

3 Answers
Jun 20, 2018

expand the bracket

#ax+ab=bx-c#

make #x# the subject

#ax+ab+c=bx#

#ab+c=bx-ax#

#ab+c=x(b-a)#

#(ab+c)/(b-a)=x#

Jun 20, 2018

#x = (c + ab)/(b - a)#

Explanation:

We have,

#color(white)(xxx)a(x + b) = bx - c#

#rArr ax + ab = bx -c# [Distributive Property]

#rArr ax +ab - bx = cancel(bx) cancel(- bx) - c# [Subtract #bx# from both sides]

#rArr ax - bx + cancel(ab) cancel(-ab) = -c - ab#

#rArr x(a- b) = (-c - ab)# [Group like terms]

#rArr x = (c + ab)/-(a - b)# [Divide both sides by #(a - b)#] [Only if #a != b#]

#rArr x = (c + ab)/(b - a)#

If #a = b#, and #ab + c != 0#, Then, The solution doesn't exist.

If #c lt 0#, #a = b# and #ab + c = 0#, Then #x in RR#

And If, #c gt 0#, #a = b# and #ab + c = 0#, Then Again, Solution is a Complex Number, so No Real Solutions.

And Gotcha.

Hope this helps.

Jun 20, 2018

If #a=b# and #ab+c=0# then we get all real numbers as solution.

if #a=b# and #ab+cne0# then no solution
if #ane b# then #x=(ab+c)/(b-a)#

Explanation:

Write your equation in the form

#x(b-a)=ab+c#
and distinguish the cases above.