First Principles AI

Solution

Using Cramer’s rule for $AX=B$ (with $|A|\ne 0$ ):

x=\frac{|A_x|}{|A|},\qquad y=\frac{|A_y|}{|A|},\qquad z=\frac{|A_z|}{|A|}

\begin{cases} 2x+y-z=1\\ x-y+2z=3\\ 3x+2y+z=4 \end{cases}

Solution:

\boxed{x=1,\; y=0,\; z=1}

\begin{cases} x_1+2x_2-3x_3=0\\ 4x_1-x_2+x_3=5\\ -2x_1+3x_2+2x_3=3 \end{cases}

Solution:

\boxed{x_1=\frac{68}{55},\; x_2=\frac{59}{55},\; x_3=\frac{62}{55}}

\begin{cases} 2x_1-x_2+x_3=1\\ x_1+2x_2+2x_3=2\\ x_1-2x_2-x_3=1 \end{cases}

Solution:

\boxed{x_1=\frac{8}{3},\; x_2=2,\; x_3=-\frac{7}{3}}