To determine which graph shows that the linear system
- \(3x + 2y = 12\)
- \(6x + 4y = 24\)
has an infinite number of solutions, we first need to analyze the two equations.
The second equation \(6x + 4y = 24\) can be simplified by dividing all terms by 2:
\[ 3x + 2y = 12 \]
This shows that both equations are actually the same, meaning they represent the same line. Since they are identical, there are infinitely many solutions (any point on this line is a solution to both equations).
Thus, a graph that displays this situation will show a single line (because the two equations are the same) or two overlapping lines representing the same equation.
From the provided options, the following description corresponds to our requirement for representing infinite solutions:
- A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two parallel lines are plotted on the plane. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma negative 6 right parenthesis. A solid downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 4 comma 0 right parenthesis.
This description does not represent our equations as they are not the same line.
None of the descriptions directly match; however, the key indication is that the two lines must be the same or one must be on top of the other to demonstrate infinite solutions.
Assuming the closest option is available, it should reflect two identical lines, not parallel or intersecting unless they overlap. Therefore based on the properties of the equations, check for the description that involves the same line mentioned earlier or matching coordinates.
If "Two intersecting lines are plotted" or "Two parallel lines" and one looks identical, that will also clarify the answer needed.
The answer could be provided or referred to "Two identical lines are plotted on the plane," but if not available, please select the option that reflects them as overlapping or indicating a singular solution visually.