Question

como resolver?
x+my=n
2x-4y=-6

Answer 1

$\begin{cases}\alpha_{1}x+\beta_{1}y=\gamma_{1}\\\alpha_{2}x+\beta_{2}y=\gamma_{2}\end{cases}$

Para sistemas lineares 2x2, podemos dizer que:

Se o sistema é S.P.D:

$\dfrac{\alpha_{1}}{\alpha_{2}}\neq\dfrac{\beta_{1}}{\beta_{2}}$

Se o sistema é S.P.I:

$\dfrac{\alpha_{1}}{\alpha_{2}}=\dfrac{\beta_{1}}{\beta_{2}}=\dfrac{\gamma_{1}}{\gamma_{2}}$

Se o sistema é S.I:

$\dfrac{\alpha_{1}}{\alpha_{2}}=\dfrac{\beta_{1}}{\beta_{2}}\neq\dfrac{\gamma_{1}}{\gamma_{2}}$
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$\begin{cases}x+my=n\\2x-4y=-6\end{cases}$

O sistema será possível e determinado (SPD) se:

$\dfrac{1}{2}\neq\dfrac{m}{-4}\\\\\\\dfrac{1}{1}\neq\dfrac{m}{-2}\\\\\\\boxed{\boxed{m\neq-2}}$
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O sistema será possível e indeterminado (SPI) se:

$\dfrac{1}{2}=\dfrac{m}{-4}=\dfrac{n}{-6}\\\\\\-\dfrac{1}{1}=\dfrac{m}{2}=\dfrac{n}{3}\\\\\\-1=\dfrac{m}{2}=\dfrac{n}{3}$

Logo:

$\dfrac{m}{2}=-1~~~\therefore~~~m=-2\\\\\\\dfrac{n}{3}=-1~~~\therefore~~~n=-3$
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O sistema será impossível (SI) se:

$\dfrac{1}{2}=\dfrac{m}{-4}\neq\dfrac{n}{-6}\\\\\\-1=\dfrac{m}{2}\neq\dfrac{n}{3}$

Logo:

$\dfrac{m}{2}=-1~~~\therefore~~~m=-2\\\\\\\dfrac{n}{3}\neq-1~~~\therefore~~~n\neq-3$