Question

Graph each system and determine number of solutions that it has. If it has one solution, name it.

$\begin{array}{c}\mathbf{x}\mathbf{-}\mathbf{y}\mathbf{=}\mathbf{2}\\ \mathbf{3}\mathbf{y}\mathbf{+}\mathbf{2}\mathbf{x}\mathbf{=}\mathbf{9}\end{array}$

Short answer

The graph of the given system of equations is:

The number of solutions that the given system of equations has is one.

The solution is $\mathbf{(}\mathbf{3}\mathbf{,}\mathbf{1}\mathbf{)}$.

Step-by-step solution

Step 1. Graph the given system of equations.

The given system of equations is:

$\begin{array}{c}x-y=2\\ 3y+2x=9\end{array}$

The equation is $x-y=2$:

Substitute 0 for $x$and find the value of localid="1647759555785" $y$.

localid="1647759424950" $\begin{array}{c}x-y=2\\ 0-y=2\\ -y=2\\ y=-2\end{array}$

Therefore, one of the point is $(0,-2)$

Substitute 0 forlocalid="1647760037493" $y$and find the value of localid="1647760031637" $x$.

$\begin{array}{c}x-y=2\\ x-0=2\\ x=2\end{array}$

Therefore, the other point islocalid="1647760002323" $(2,0)$

Therefore, the graph of the equationlocalid="1647760006510" $x-y=2$is the line passing through the pointslocalid="1647760010950" $(0,-2)$and localid="1647760014937" $(2,0)$.

The equation is localid="1647759996770" $3y+2x=9$:

Substitute 0 forlocalid="1647760043865" $x$and find the value of localid="1647760025133" $y$.

localid="1647760018807" $\begin{array}{c}3y+2x=9\\ 3y+2\left(0\right)=9\\ 3y=9\\ y=3\end{array}$

Therefore, one of the point is localid="1647760053857" $(0,3)$

Substitute 0 forlocalid="1647760065676" $y$and find the value of localid="1647760058997" $x$.

localid="1647760069632" $\begin{array}{c}3y+2x=9\\ 3\left(0\right)+2x=9\\ 2x=9\\ x=4.5\end{array}$

Therefore, the other point islocalid="1647760077077" $(4.5,0)$

Therefore, the graph of the equation localid="1647760083440" $3y+2x=9$is the line passing through the points localid="1647759991807" $(0,3)$and localid="1647760089032" $(4.5,0)$.

Therefore, the graph of the given system of equations is:

Step 2. Determine the number of solutions that the given system of equations has.

From the graph of the system of equations, it can be noticed that the intersection point of the lines is at$(3,1)$ and the number of intersection point is 1.

Therefore, the given system of equations has one solution and the solution is $(3,1)$.