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Which of these graphs shows that the linear system −2x+y=6

and 4x−2y=−12
has an infinite number of solutions?(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 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. 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.
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 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.

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. 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.

1 answer

For the linear system to have an infinite number of solutions, the two equations must represent the same line, meaning they are dependent and coincide at every point.

Let's analyze the equations given:

  1. The first equation is \( -2x + y = 6 \).
  2. The second equation is \( 4x - 2y = -12 \).

We can manipulate the second equation to see if it can be transformed into the first:

Starting with \( 4x - 2y = -12 \), we divide every term by 2: \[ 2x - y = -6 \]

Rearranging gives us: \[ -y = -2x - 6 \quad \Rightarrow \quad y = 2x + 6 \]

The two equations can be rewritten as:

  1. \( y = 2x + 6 \)
  2. \( y = 2x + 6 \) (the same equation)

Since both equations describe the same line, the system has an infinite number of solutions.

From the descriptions provided, the graph that shows the two equations has to be a single line. Since there’s no parallel line and no intersecting line (because intersecting lines would indicate a single solution), the graph that shows the infinite solutions must represent a single line.

Thus, the response that indicates the correct graph is the one that shows a single line passing through points. Therefore:

  • Answer: The first graph you described, which shows a single upward slanting line passing through the points (0, -6) and (3, 0), represents the situation correctly.