Technique for solving \(\displaystyle{\frac{{{a}{x}+{b}}}{{{c}{x}+{d}}}}={\frac{{{p}{x}+{q}}}{{{r}{x}+{s}}}}\) where the sum

Zoie Phillips

Zoie Phillips

Answered question

2022-04-01

Technique for solving ax+bcx+d=px+qrx+s where the sum of numerators equals the sum of denominators
I was looking up some shortcuts to solve quadratic equations. I got a technique that applies when the sum of the numerators and denominators are equal, but I am unable to understand the reasoning behind it. Here I'm showing an example:
3x+46x+7=5x+62x+3

Answer & Explanation

ron4d3ozip7

ron4d3ozip7

Beginner2022-04-02Added 9 answers

Your equation has the form
N1D1=N2D2
that implies N1D2=N2D1
Add N2D2 to both sides,
(N1+N2)D2=N2(D1+D2)()
but (N1+N2)=(D1+D2), so you cam simplify the terms in the parenthesis, and remain with
D2=N2D2N2=0
Similarly you can show that an equivalent condition is D1N1=0.
Note that equation (∗) is satisfied also if (N1+N2)=(D1+D2)=0, which gives the other solution.

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