Which of these graphs shows that the linear system 3x+2y=12

Question

and 6x+4y=24
has an infinite number of solutions? graph 1,2,3,or 4?(1 point)
Responses

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. A 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.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. A 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.

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.
Image with alt text: 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.

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two intersecting lines are plotted on the plane. A solid downward slanting line passes through the points left parenthesis negative 6 comma 3 right parenthesis and left parenthesis negative 4 comma 0 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 6 comma 4 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. Two intersecting lines are plotted on the plane. A solid downward slanting line passes through the points left parenthesis negative 6 comma 3 right parenthesis and left parenthesis negative 4 comma 0 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 6 comma 4 right parenthesis.

A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. A downward slanting line with arrows at both ends passes through the points left parenthesis negative 10 comma 9 right parenthesis and left parenthesis negative 4 comma 0 right parenthesis.
Image with alt text: A coordinate plane with 4 quadrants shows x and y axes ranging from negative 10 to 10 in unit increments. A line is plotted on the plane. A downward slanting line with arrows at both ends passes through the points left parenthesis negative 10 comma 9 right parenthesis and left parenthesis negative 4 comma 0 right parenthesis.
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Answer 1

To determine which graph represents the linear system having an infinite number of solutions, we should consider the nature of the equations given:

1. The two equations are:

   • \(3x + 2y = 12\)
   • \(6x + 4y = 24\)

2. Notice that the second equation is simply a multiple of the first equation (i.e., it can be obtained by multiplying the entire first equation by 2). This means that the two lines represented by these equations are the same line, and therefore the system has an infinite number of solutions (every point on the line is a solution).

Now, looking at the descriptions of the graphs:

• Graph 1: Represents a single line.
• Graph 2: Represents two parallel lines (not the same; hence finite solutions).
• Graph 3: Represents two intersecting lines (which would have one solution).
• Graph 4: Represents a single line.

The correct graph that shows the linear system has an infinite number of solutions is Graph 2, as it shows the same line represented in two ways which indicates they are equivalent, fulfilling the criteria of having infinite solutions.