Question

For each system, choose the best description of its solution.If applicable, give the solution.The system has no solution.The system has a unique solution:(x, y) = 0.0System Ax+3y+9 = 0-x-3y=-9The system has infinitely many solutions.They must satisfy the following equation:p= 0The system has no solution.System BThe system has a unique solution:(x, y) = 0.0x-4y=-8-x-4y= -8The system has infinitely many solutions,They must satisfy the following equation:

Answer 1

(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:

`\begin{gathered} x+3y+9=0 \\ -x-3y=-9 \end{gathered}`

Rewrite both equations in the slope-intercept form:

`\begin{gathered} Equation\; 1\colon3y=-x-9$$\textcolor{red}{\implies y=-(x)/(3)-(9)/(3)}$$ \\ Equation\; 2\colon3y=-x+9\textcolor{red}{\implies y=-(x)/(3)+(9)/(3)} \end{gathered}`

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:

`\begin{gathered} x-4y=-8 \\ -x-4y=-8 \end{gathered}`

Add both equations:

`-8y=-16`

Divide both sides by -8.

`\begin{gathered} (-8y)/(-8)=(-16)/(-8) \\ y=2 \end{gathered}`

Substitute y=2 into the first equation to solve for x.

`\begin{gathered} x-4y=-8 \\ x-4(2)=-8 \\ x-8=-8 \\ x=-8+8 \\ x=0 \end{gathered}`

The system has a unique solution (x,y) = (0,2).