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# Ratio and Propotion

$$m,~ n,~p,~q,~are~real ~numbers~,p~and~q~ both~ together~ not ~equal~to~zero. \\ If~~\dfrac {a}{b} =\dfrac {c}{d} \\ \implies \dfrac {a}{b} =\dfrac {c}{d}=\dfrac {pa+qc}{pb+qd}.\\Also\dfrac {ma+nb}{pa+qb} =\dfrac {mc+nd}{pc+qd}$$
This can be applied any where with a, b, c, d, are any expression. We can apply it in solving simultaneous equations in two variables. It is similar to elimination method.
$$2a - 5b = 11$$
$$3a+2b = 7$$
$$\implies \dfrac{2a - 5b }{3a+2b } = \dfrac{11}{7}.~~~~\\$$
Subtract numerator from denominator on both the sides.
$$\implies \dfrac{2a - 5b }{a+7b } = \dfrac{11}{-4}$$

Subtract twice the denominator from the numerator on both the sides.
$$\implies \dfrac{-19b}{a+7b } = \dfrac{19}{-4}~\\$$
$$\implies \dfrac{b }{a+7b } = \dfrac{1}{4}~~~~~~~~~~~~\\$$
Subtract seven times the numerator from denominator on both the sides.

$$\implies \dfrac{b }{a} = \dfrac{1}{-3}$$ $$\implies a=-3b~\\$$

Substitute in any one equation and we get ..... a=3~~~b=-1

Note by Niranjan Khanderia
2 years, 5 months ago

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i think above method is lengthy for such a trivial problem of simultaneous equations · 8 months, 2 weeks ago