Question

(UEL-PR) Numa loja, os artigos A e B, juntos, custam R$70,00, dois artigos A mais
um artigo C custam R$105,00 e a diferença de preço entre os artigos B e C, nessa
ordem, é R$5,00. Qual é o preço do artigo C?

Como solucionar usando a fórmula de Cramer

Answer 1

Vamos começar montando o sistema:

$\left\{\begin{array}{ccccc}A&+~~~B&~&=&70\\2A&~&+~~~C&=&105\\~&B&-~~~C&=&5\end{array}\right$

Vamos, agora, determinar a matriz incompleta do sistema (matriz sem os termos independentes):

$A~=~\left[\begin{array}{ccccc}1&1&0\\2&0&1\\0&1&-1\end{array}\right]\\\\\\Calculando~o~seu~determinante:\\\\\\det(A)~=~(~1.0.(-1)+2.1.0+0.1.1~)~-~(~0.0.0+1.1.1+(-1).1.2~)\\\\\\det(A)~=~(~0+0+0~)~-~(~0+1-2~)\\\\\\det(A)~=~0~-~(-1)\\\\\\\boxed{det(A)~=~1}$

Podemos agora montar a matriz Ax, substituindo a primeira coluna da matriz A pela coluna de termos independentes:

$Ax~=~\left[\begin{array}{ccccc}70&1&0\\105&0&1\\5&1&-1\end{array}\right]\\\\\\Calculando~o~seu~determinante:\\\\\\det(Ax)~=~(~70.0.(-1)+105.1.0+5.1.1~)~-~(~0.0.5+1.1.70+(-1).1.105~)\\\\\\det(Ax)~=~(~0+0+5~)~-~(~0+70-105~)\\\\\\det(Ax)~=~(~5~)~-~(~-35~)\\\\\\\boxed{det(Ax)~=~40}$

Podemos agora montar a matriz Ay, substituindo a segunda coluna da matriz A pela coluna de termos independentes:

$Ay~=~\left[\begin{array}{ccccc}1&70&0\\2&105&1\\0&5&-1\end{array}\right]\\\\\\Calculando~o~seu~determinante:\\\\\\det(Ay)~=~(~1.105.(-1)+2.5.0+0.70.1~)~-~(~0.105.0+1.5.1+(-1).70.2~)\\\\\\det(Ay)~=~(~-105+0+0~)~-~(~0+5-140~)\\\\\\det(Ay)~=~(~-105~)~-~(~-135~)\\\\\\\boxed{det(Ay)~=~30}$

Podemos agora montar a matriz Az, substituindo a terceira coluna da matriz A pela coluna de termos independentes:

$Az~=~\left[\begin{array}{ccccc}1&1&70\\2&0&105\\0&1&5\end{array}\right]\\\\\\Calculando~o~seu~determinante:\\\\\\det(Az)~=~(~1.0.5+2.1.70+0.1.105~)~-~(~70.0.0+105.1.1+5.1.2~)\\\\\\det(Az)~=~(~0+140+0~)~-~(~0+105+10~)\\\\\\det(Az)~=~(~140~)~-~(~115~)\\\\\\\boxed{det(Az)~=~25}$

Podemos agora calcular o valor dos artigos A, B e C, acompanhe:

$A~=~\frac{det(Ax)}{det(A)}~=~\frac{40}{1}~~\rightarrow~~\boxed{A~=~R\ \,40,00}\\\\\\B~=~\frac{det(Ay)}{det(A)}~=~\frac{30}{1}~~\rightarrow~~\boxed{B~=~R\ \,30,00}\\\\\\C~=~\frac{det(Az)}{det(A)}~=~\frac{25}{1}~~\rightarrow~~\boxed{C~=~R\ \,25,00}$