Question

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

$\begin{array}{l}3x+y=5\\ 6x=10-2y\end{array}$

Short answer

The graph of the system is

The number of solutions is infinitely many.

Step-by-step solution

Step 1. Graph the system of equations.

The given system of equation is:

$3x+y=5$ … (1)

$6x=10-2y$ … (2)

The graph of two equations is drawn as below.

Step 2. Interpret the system of equations and graph.

Rewrite the given system of equation:

$$\begin{gathered} 3x+y=5 \\ y=-3x+5\dots \left(1\right) \end{gathered}$$

$$\begin{gathered} 6x=10-2y \\ 2y=-6x+10 \\ y=-3x+5\dots \left(2\right) \end{gathered}$$

The two equations represented eq. (1) and (2) are same. That’s why there is a single line in graph of two equations as they are overlapping with each other.

Step 3. Name the solutions, if possible.

When a graph of two equations gives a same line of graph, it contains the infinite number of solutions.

Therefore, infinite number of solutions is observed for given system of equation and hence naming all the solutions is not possible.