First let's find the slope and y-intercept of each equation by putting them in the form y = mx + b:
[tex]\begin{gathered} \frac{1}{3}x+y=7 \\ y=-\frac{1}{3}x+7 \\ \text{slope}\to m=-\frac{1}{3} \\ y-\text{intercept}\to b=7 \\ \\ 2x+6y=12 \\ x+3y=6 \\ 3y=-x+6 \\ y=-\frac{1}{3}x+2 \\ \text{slope}\to m=-\frac{1}{3} \\ y-\text{intercept}\to b=2 \end{gathered}[/tex]Since the lines have the same slope and different y-intercepts, they are parallel lines, therefore these lines have no point of intersection.
