Question

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

$\begin{array}{l}\mathbf{y}\mathbf{=}\mathbf{2}\mathbf{x}\\ \mathbf{y}\mathbf{=}\mathbf{6}\mathbf{-}\mathbf{x}\end{array}$

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 $\mathbf{(}\mathbf{2}\mathbf{,}\mathbf{4}\mathbf{)}$.

Step-by-step solution

Step 1. Graph the given system of equations.

The given system of equations is:

$\begin{array}{l}y=2x\\ y=6-x\end{array}$

The equation is:$y=2x$

Substitute 0 for$x$and find the value of$y$.

$\begin{array}{l}y=2x\\ y=2\left(0\right)\\ y=0\end{array}$

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

Substitute 1 for$x$and find the value of$y$.

$\begin{array}{l}y=2x\\ y=2\left(1\right)\\ y=2\end{array}$

Therefore, the other point is$(1,2)$

Therefore, the graph of the equation$y=2x$is the line passing through the points$(0,0)$and$(1,2)$.

The equation is:$y=6-x$

Substitute 0 for$x$and find the value of$y$.

$\begin{array}{l}y=6-x\\ y=6-0\\ y=6\end{array}$

Therefore, one of the point is$(0,6)$

Substitute 0 for$y$and find the value of$x$.

$\begin{array}{l}y=6-x\\ 0=6-x\\ x=6\end{array}$

Therefore, the other point is$(6,0)$

Therefore, the graph of the equation$y=6-x$is the line passing through the points$(0,6)$and$(6,0)$.

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$(2,4)$ and the number of intersection point is 1.

Therefore, the given system of equations has one solution and the solution is $(2,4)$.