Which of these graphs shows that the linear system −2x+y=6 and 4x−2y=−12 has an infinite number of solutions?(1 point)

Question

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 0 comma negative 6 right parenthesis and left parenthesis 3 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 3 comma 0 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 negative 3 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 3 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. 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 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 1 comma 7 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 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A dotted upward slanting line passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 1 comma 7 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 upward slanting line with arrows at both ends passes through the points left parenthesis negative 3 comma 0 right parenthesis and left parenthesis 0 comma 6 right parenthesis. A solid upward slanting line with arrows at both ends, parallel to the dotted line, passes through origin.

Answer 1

To determine whether the linear system \(-2x + y = 6\) and \(4x - 2y = -12\) has an infinite number of solutions, we need to check if the two equations represent the same line. This occurs if one equation can be derived from the other by multiplying or dividing by a non-zero constant, leading to the two equations being equivalent.

Let's first manipulate the second equation:

1. Start with \(4x - 2y = -12\).

2. Divide the entire equation by 2:

   \[ 2x - y = -6. \]

3. Rearranging gives:

   \[ y = 2x + 6. \]

Now we compare this with the first equation \(-2x + y = 6\). Rearranging the first equation gives us:

\[ y = 2x + 6. \]

Both equations simplify to the same line, \(y = 2x + 6\).

From this analysis, we determine that both equations describe the same line, thus indicating that the linear system has an infinite number of solutions.

The graph that shows this is the one with two parallel lines. This is because if the equations were the same line, there would be one unique line. Instead, if they were represented in different forms but remained parallel (and not intersecting), that would suggest multiple solutions as well.

Based on the given options, you need to look for a graph that shows two parallel lines, which would indicate the system has an infinite number of solutions.

The fourth graph option, which indicates the presence of two parallel lines, is the correct answer.