Question

Resolva os sistemas lineares pela regra de Cramer:

A) 2X + 3Y = 13

3X – 5Y = 10


B) 3X + Y = 9

2X + 3Y = 13​

Answer 1

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$\boxed{\begin{array}{l}\rm a)\\\begin{cases}\sf 2x+3y=13\\\sf 3x-5y=10\end{cases}\\\sf A=\begin{bmatrix}\sf2&\sf3\\\sf3&\sf-5\end{bmatrix}\\\sf det~A=2\cdot(-5)-3\cdot3\\\sf det~A=-10-9\\\sf det~A=-19\\\sf A_x=\begin{bmatrix}\sf13&\sf3\\\sf10&\sf-5\end{bmatrix}\\\sf det~A_x=13\cdot(-5)-3\cdot10\\\sf det~A_x=-65-30\\\sf det~A_x=-95\\\sf A_y=\begin{bmatrix}\sf2&\sf13\\\sf3&\sf10\end{bmatrix}\\\sf det~A_y=2\cdot10-3\cdot13\\\sf det~A_y=20-39\\\sf det~A_y=-19\end{array}}$

$\large\boxed{\begin{array}{l}\sf x=\dfrac{det~A_x}{det~A}\\\\\sf x=\dfrac{-95}{-19}\\\sf x=5\\\sf y=\dfrac{det~A_y}{det~A}\\\\\sf y=\dfrac{-19}{-19}\\\sf y=1\\\sf S=\{5,1\}\end{array}}$

$\boxed{\begin{array}{l}\rm b)\\\begin{cases}\sf 3x+y=9\\\sf 2x+3y=13\end{cases}\\\sf A=\begin{bmatrix}\sf3&\sf1\\\sf2&\sf3\end{bmatrix}\\\sf det~A=3\cdot3-2\cdot1\\\sf det~A=9-2\\\sf det~A=7\\\sf A_x=\begin{bmatrix}\sf9&\sf1\\\sf13&\sf3\end{bmatrix}\\\sf det~A_x=9\cdot3-13\cdot1\\\sf det~A_x=27-13\\\sf det~A_x=14\\\sf A_y=\begin{bmatrix}\sf3&\sf9\\\sf2&\sf13\end{bmatrix}\\\sf det~A_y=3\cdot13-2\cdot9\\\sf det~A_y=39-18\\\sf det~A_y=21\end{array}}$

$\Large\boxed{\begin{array}{l}\sf x=\dfrac{det~A_x}{det~A}\\\\\sf x=\dfrac{14}{7}\\\\\sf x=2\\\\\sf y=\dfrac{det~A_y}{det~A}\\\\\sf y=\dfrac{21}{7}\\\\\sf y=3\\\sf S=\{2,3\}\end{array}}$