Answer:
(A)System A has no solution
(B)System B has a unique solution (x,y) = (0,2).
Explanation:
System A
Given the system of equations:
[tex]\begin{gathered} x+3y+9=0 \\ -x-3y=-9 \end{gathered}[/tex]Rewrite both equations in the slope-intercept form:
[tex]\begin{gathered} Equation\; 1\colon3y=-x-9$\textcolor{red}{\implies y=-\frac{x}{3}-\frac{9}{3}}$ \\ Equation\; 2\colon3y=-x+9\textcolor{red}{\implies y=-\frac{x}{3}+\frac{9}{3}} \end{gathered}[/tex]On observation, the slopes of the two equations = -1/3.
This means that the two lines are parallel and thus, the system has no solution.
System B
Given the system of equations:
[tex]\begin{gathered} x-4y=-8 \\ -x-4y=-8 \end{gathered}[/tex]Add both equations:
[tex]-8y=-16[/tex]Divide both sides by -8.
[tex]\begin{gathered} \frac{-8y}{-8}=\frac{-16}{-8} \\ y=2 \end{gathered}[/tex]Substitute y=2 into the first equation to solve for x.
[tex]\begin{gathered} x-4y=-8 \\ x-4(2)=-8 \\ x-8=-8 \\ x=-8+8 \\ x=0 \end{gathered}[/tex]The system has a unique solution (x,y) = (0,2).
