System A
[tex]\begin{gathered} x-4y=4 \\ -x+4y+4=0 \end{gathered}[/tex]
solve the first equation for x
[tex]x=4+4y[/tex]
replace in the second equation
[tex]\begin{gathered} -(4+4y)+4y+4=0 \\ -4-4y+4y+4=0 \\ 0=0 \end{gathered}[/tex]
The system has infinitely many solutions, They must satisfy the following equation
[tex]\begin{gathered} x-4y=4 \\ -4y=4-x \\ y=\frac{4}{-4}-\frac{x}{-4} \\ y=\frac{x}{4}-1 \end{gathered}[/tex]
System B
[tex]\begin{gathered} x-2y=6 \\ -x+2y=6 \end{gathered}[/tex]
solve for x for the first equation
[tex]x=6+2y[/tex]
replace in the second equation
[tex]\begin{gathered} -(6+2y)+2y=6 \\ -6-2y+2y=6 \\ -6=6 \end{gathered}[/tex]
The system has no solution.
