Question

Utilizando a regra de Cramer, determine o valor da incógnita y no seguinte sistema de equações lineares:?
2x+3y+3z=18
3x+2y+5z=23
5x+4y+2z=27

Answer 1

Oi 
Calculando o Determinante da matriz principal (retirando as variáveis x,y,z):

$D= \left[\begin{array}{ccc}2&3&3\\3&2&5\\5&4&2\end{array}\right] \\ \\ D=(2.2.2)+(3.5.5)+(3.3.4)-[(3.2.5)+(5.4.2)+(2.3.3)] \\ \\ D=8+75+36-[30+40+18] \\ \\ D=119-81 \\ \\ D=31$

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Calculando Dx :

$D_x= \left[\begin{array}{ccc}18&3&3\\23&2&5\\27&4&2\end{array}\right] \\ \\ D_x=(18.2.2)+(3.5.27)+(3.23.4)-[(3.2.27)+(5.4.18)+(2.3.23)] \\ \\ D_x=72+405+276-[162+360+138] \\ \\ D_x=753-660 \\ \\ \boxed{D_x=93}$

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Calculando Dy:

$D_y= \left[\begin{array}{ccc}2&18&3\\3&23&5\\5&27&2\end{array}\right] \\ \\ D_y=(2.23.2)+(18.5.5)+(3.3.27)-[(3.23.5)+(5.27.2)+(2.3.18)] \\ \\ D_y=92+450+243-[345+270+108] \\ \\ D_y=785-723 \\ \\ \boxed{D_y=62}$


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Calculando Dz:

$D_z= \left[\begin{array}{ccc}2&3&18\\3&2&23\\5&4&27\end{array}\right] \\ \\ D_z=(2.2.27)+(3.23.5)+(18.3.4)-[(18.2.5)+(23.4.2)+(27.3.3)] \\ \\ D_z=108+345+216-[180+184+243] \\ \\ D_z=669-607 \\ \\ \boxed{D_z=62}$

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Encontrado os determinantes . Agora é só encontrar o valor dos coeficientes: 

$x= \frac{D_x}{D} = \frac{93}{31} =3 \\ \\ y= \frac{D_y}{D} = \frac{62}{31} =2 \\ \\ z= \frac{D_z}{D} = \frac{62}{31} =2$

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Portanto:   

x=3  , y=2 , z=2  

Espero que goste. Comenta Depois :)