Question

calcule.

necessito
dado o sistema abaixo,resolva aplicando a regra de crammer.
​

Answer 1

Os valores das incógnitas são: x = 62, y = 45 e z = 29.

• Resolvendo o problema

Para aplicar a Regra de Cramer, primeiro montamos a matriz M com os coeficientes das equações:

$M=\left[\begin{array}{ccc}1&1&0\\0&1&1\\1&0&1\end{array}\right]$

Agora, calculamos o seu determinante

$D=(1\;.\;1\;.\;1)+(1\;.\;1\;.\;1)+(0\;.\;0\;.\;0)-(0\;.\;1\;.\;1)-(1\;.\;1\;.\;0)-(1\;.\;0\;.\;1)\\\\D=1+1+0-0-0-0\\\\D=2$

Agora calculamos os determinantes relativos a cada incógnita substituindo, na matriz M, a coluna correspondente pelos termos independentes.

$M_x=\left[\begin{array}{ccc}107&1&0\\74&1&1\\91&0&1\end{array}\right]\\\\\\D_x=(107\;.\;1\;.\;1)+(1\;.\;1\;.\;91)+(0\;.\;74\;.\;0)-(0\;.\;1\;.\;91)-(107\;.\;1\;.\;0)-(1\;.\;74\;.\;1)\\\\D_x=107+91+0-0-0-74\\\\D_x=124$

$M_y=\left[\begin{array}{ccc}1&107&0\\0&74&1\\1&91&1\end{array}\right] \\\\\\D_y=(1\;.\;74\;.\;1)+(107\;.\;1\;.\;1)+(0\;.\;0\;.\;91)-(0\;.\;74\;.\;1)-(1\;.\;1\;.\;91)-(107\;.\;0\;.\;1)\\\\D_y=74+107+0-0-91-0\\\\D_y=90$

$M_z=\left[\begin{array}{ccc}1&1&107\\0&1&74\\1&0&91\end{array}\right]\\\\\\D_z=(1\;.\;1\;.\;91)+(1\;.\;74\;.\;1)+(107\;.\;0\;.\;0)-(107\;.\;1\;.\;1)-(1\;.\;74\;.\;0)-(1\;.\;0\;.\;91)\\\\D_z=91+74+0-107-0-0\\\\D_z=58$

Agora, basta encontrar as incógnitas dividindo o determinante correspondente pelo determinante de M.

$x=\dfrac{D_x}{D}=\dfrac{124}{2}=62\\\\\\y=\dfrac{D_y}{D}=\dfrac{90}{2}=45\\\\\\z=\dfrac{D_z}{D}=\dfrac{58}{2}=29$

• Conclusão

Portanto, os valores das incógnitas são: x = 62, y = 45 e z = 29.

• Para saber mais

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