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. An upward slanting line with arrows at both ends passes through the points left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma 6 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. An upward slanting line with arrows at both ends passes through the points left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma 6 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. An upward slanting line with arrows at both ends passes through the points left parenthesis 0 comma negative 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. An upward slanting line with arrows at both ends passes through the points left parenthesis 0 comma negative 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 upward slanting line with arrows at both ends passes through the points left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 2 comma 3 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 upward slanting line with arrows at both ends passes through the points left parenthesis negative 4 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 2 comma 3 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 solid upward slanting line with arrows at both ends passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 4 comma 0 right parenthesis. A dotted upward slanting line with arrows at both ends passes through the points left parenthesis 6 comma 7 right parenthesis and left parenthesis 8 comma 10 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 solid upward slanting line with arrows at both ends passes through the points left parenthesis 0 comma negative 6 right parenthesis and left parenthesis 4 comma 0 right parenthesis. A dotted upward slanting line with arrows at both ends passes through the points left parenthesis 6 comma 7 right parenthesis and left parenthesis 8 comma 10 right parenthesis.
Remaining Attempts : 3
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Answer 1

To determine which graph shows that the linear system given by the equations \(3x - 2y = 12\) and \(-6x + 4y = -24\) has an infinite number of solutions, we need to analyze the two equations.

First, we can simplify the second equation:

\[ -6x + 4y = -24 \]

Dividing the entire equation by -2 gives:

\[ 3x - 2y = 12 \]

Notice that after simplification, both equations \(3x - 2y = 12\) and \(-6x + 4y = -24\) are equivalent. This means that they represent the same line.

When two linear equations represent the same line on a graph, it means that they have an infinite number of solutions since every point on that line is a solution to both equations.

Now, looking at the descriptions of the graphs:

1. Graph 1: This shows one line.
2. Graph 2: This shows one line.
3. Graph 3: This shows two intersecting lines.
4. Graph 4: This shows two parallel lines.

The correct answer is Graph 1 or Graph 2—both can represent the same line, leading to an infinite number of solutions. However, since the options point towards two distinct representations of a single line and not two lines, we typically look for identical overlapping lines which are presented in the format of Graph 1 or Graph 2.

If you need to pick one, since both Graph 1 and Graph 2 represent a single line each, you can choose either of these based on their described characteristics—just ensure they actually represent the same line as derived.

Thus, please confirm the visuals that match the identical equations (i.e., overlapping lines). In abundant context, choose either Graph 1 or Graph 2.