To determine the nature of the graph of the given system of equations, let's analyze the two equations.
The equations are:
[tex]\[
\begin{cases}
2x + y = 6 \\
6x + 3y = 12
\end{cases}
\][/tex]
First, we simplify the second equation by dividing every term by 3:
[tex]\[
6x + 3y = 12 \implies 2x + y = 4
\][/tex]
Now we have the simplified system:
[tex]\[
\begin{cases}
2x + y = 6 \\
2x + y = 4
\end{cases}
\][/tex]
This simplified system clearly shows that the two equations have the same coefficients for x and y, but different constants on the right-hand side (6 and 4).
When two linear equations have the same left-hand side (coefficients for x and y) but different right-hand sides, they represent parallel lines. Parallel lines have the same slope but different intercepts, which means they never intersect.
Therefore, the graph of this system of equations is:
Parallel lines
