Solve for x, a/x+a + b/x-b = (a+b)/(x+c)

Question

Year 11 Math Methods

Answer 1

Simplify then solve:

 

(ax-ba+bx+ab)/(x+a)(x-b)=(a+b)/(x+c)

 

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

 

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

 

x² +cx = x² +ax-bx-ab

 

ax -bx -cx =ab

 

(a-b-c) x = ab

 

x = ab/(a-b-c)

 

 

Answer 2

Solve for x, a/x+a + b/x-b = (a+b)/(x+c)

 

This one is a bit too messy to type out without a real equation editor, but here's how to proceed.

 

(I assumed that the x+a and x-b should have been in parentheses.)

 

Find a common denominator for all three fractions.  That will be (x+a)(x-b)(x+c).

The numerators of the fractions will become (a)(x-b)(x+c), (b)(x+a)(x+c) and (x+a)(x-b)(x+c)

Subtract the big radical expression on the right from both sides so the right side becomes zero.

Now you can expand all of the numerators add the fractions, and combine like terms.

Multiply both sides by the denominator to eliminate it.  (The right side is zero!  See Domain issues below.)

I got all of the x-squared terms and all but one of the constant terms to add out.

Move the constant term to the right side.

Factor the x out of all of the x-terms leaving an x-coefficient.  I got (ac+bc+ab+b²)x =-ab²

Divide both sides by that 4-term x-coefficient to get x = ...

 

Domain issues:  Look at the original form of the equation.  x cannot be -a, +b, or -c.

 

Hope this helps.