Solve for x, a/(ax-1) + b/(bx-1) = a+b

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hala718 | High School Teacher | (Level 1) Educator Emeritus

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a/ax-1 + b/bx-1 = a+b

=> ( a- ax) / ax  + ( b-bx)/ bx = (a+ b)

We will multiply by abx^2

==> (a- ax)(b-bx) = (a+b) abx^2

==> (ab - 2abx + abx^2 = (a+b) abx^2

==> ab - 2abx = (a+b)( abx^2) - abx^2

We will facro abx^2 from the right side.

==> ab -2abx = abx^2 ( a+b -1)

We will factor ab from the left side.

==> ab ( -2x +1) = abx^2 ( a+ b-1)

Now we will divide by ab.

==> (-2x+1) = x^2 ( a+ b -1)

==> (a+b-1) x^2 +2x -1 = 0

Now we have a quadratic equation, we will solve for x in terms of a and b.

==> x1= ( -2 + sqrt(4 +4(a+b-1) / 2(a+b-1)

            = ( -2+ 2sqrt(a+b) / 2(a+b-1)

           = (-1+ sqrt(a+b) / (a+b-1)

==>x1= (-1+ sqrt(a+b) / (a+b-1)

==> x2= ( -1- sqrt(a+b) / (a+b-1)

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stutipai | Student, Grade 10 | eNotes Newbie

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a/(ax – 1) + b/(bx – 1) = ab ⇒ a – a/(ax – 1) + b – b/(bx – 1) = 0 ⇒ [a (ax – 2)(bx – 1) + b(bx –  2)(ax – 1)] / [(ax – 1) (bx – 1)] = 0 ⇒ ab (ab)x – [(ab)2 +2abx + 2 (ab) = 0 ⇒ ab (ab)abx – (ab)2x – 2abx + 2 (ab) = 0 ⇒ (ab)x[abx – (ab)] – 2[abx – (ab)] = 0 ⇒ [(ab)x – 2][abx – (ab)] = 0 ⇒ [(ab)x – 2] = 0 or [abx – (ab)] = 0 ⇒ x = 2/(ab) or x = (ab)/ab

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