Answer to Question #200206 in Linear Algebra for Shadleigh

Question

Use Cramer's rule to solve the equation below:

y − z = 2

3x + 2y + z = 4

5x + 4y =1 

Answer 1

$given\\ y − z = 2\\ 3x + 2y + z = 4\\ 5x + 4y =1 \\ we\space can\space write\space this\space as\\ 0x+y − z = 2\\ 3x + 2y + z = 4\\ 5x + 4y +0z=1 \\ Step \space 1: \space Find\space the \space determinant\space of \space the\space coefficient\space matrix.\\ \begin{vmatrix} 0 & 1 & -1\\ 3 & 2 & 1\\ 5 & 4 &0 \end{vmatrix} \\ Now\space find\space determinant \space as\space \\ D=0(2×0-1×4)+1(1×5-3×0)+(-1)(3×4-2×5)=3\\ Step2:\space Find \space the \space determinant\space of\space the\space x -\space matrix\space ( D_x ).\space X \space - \space matrix\space is \space formed\space by \space replacing\space the \space x-column\space values \space with \space the \space answer-column\space values\\ \begin{vmatrix} 2 & 1 & -1\\ 4& 2 & 1\\ 1 & 4 &0 \end{vmatrix} \\ Now \space find \space determinant \space = D_x =-21\\ Step3:\space Find\space the \space determinant\space of \space the\space y - matrix\space \space ( D_y ). \space Y - matrix\space is \space formed\space by \space replacing \space the\space y-column \space values\space with \space the \space answer-column \space values\\ \begin{vmatrix} 0 & 2& -1\\ 3 & 4 & 1\\ 5 & 1&0 \end{vmatrix} \\ Now\space find \space determinant \space = D_y= 27\\ Step4:\space Find\space the \space determinant\space of \space the\space z - matrix\space \space ( D_z ). \space Z - matrix\space is \space formed\space by \space replacing \space the\space z-column \space values\space with \space the \space answer-column \space values\\ \begin{vmatrix} 0 & 1 & 2\\ 3 & 2& 4\\ 5 & 4 &1 \end{vmatrix} \\ Now\space find \space determinant \space = D_z= 21\\ Cramers \space Rule\space says\space that\space the\space solutions \space are\\ x=\frac{D_x }{D}=\frac{-21}{3}=-7\\ y=\frac{D_y }{D}=\frac{27}{3}=9\\ z=\frac{D_z }{D}=\frac{21}{3}=7\\$