Question

Qual o valor a atribuir ao parâmetro m para que os sistemas sejam equivalentes ?

( C.N. - 1954 )

{ mx + 2my = 1
{ mx + 3my = 2

e

{ x = a
{ y = - a​

Answer 1

$\boxed{\begin{array}{l}\sf dois~sistemas~s\tilde ao~equivalentes\\\sf quando~tem~o~mesmo~conjunto~soluc_{\!\!,} \tilde ao. \end{array}}$

$\begin{bmatrix}\sf m&\sf2m\\\sf m&\sf3m\end{bmatrix}\cdot\begin{bmatrix}\sf x\\\sf y\end{bmatrix}=\begin{bmatrix}\sf1\\\sf 2\end{bmatrix}\\\sf A=\begin{bmatrix}\sf m&\sf2m\\\sf m&\sf3m\end{bmatrix}\\\sf det~A=3m^2-2m^2\\\sf det~A= m^2\\\sf A_x=\begin{bmatrix}\sf1&\sf2m\\\sf2&\sf 3m\end{bmatrix}\\\sf det~A_x=3m-4m\\\sf det~A_x=-m\\\sf A_y=\begin{bmatrix}\sf m&\sf1\\\sf m&\sf2\end{bmatrix}\\\sf det~ A_y=2m-m\\\sf det~ A_y=m\\\sf x=\dfrac{det~A_x}{det~A}\\\sf x=\dfrac{-m}{m^2}=-\dfrac{1}{m}\\\sf a=-\dfrac{1}{m}\\\sf m=-\dfrac{1}{a}, a\ne0\\\sf y=\dfrac{det~A_y}{det~A}\\\sf y=\dfrac{m}{m^2}\\\sf y=\dfrac{1}{m}\\\sf -a=\dfrac{1}{m}\\\sf m=-\dfrac{1}{a},a\ne0$