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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 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.
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. 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.

1 answer

To determine which graph shows that the linear system:

  1. \(-2x + y = 6\)
  2. \(4x - 2y = -12\)

has an infinite number of solutions, we should first check if the two equations represent the same line.

We can manipulate the first equation to get it in slope-intercept form (y = mx + b):

  1. From \(-2x + y = 6\), we can rearrange it to \(y = 2x + 6\).

Next, let's manipulate the second equation.

  1. The second equation \(4x - 2y = -12\) can be rearranged as follows:

    \[ -2y = -4x - 12 \] \[ y = 2x + 6 \]

Since both equations simplify to \(y = 2x + 6\), they represent the same line. Therefore, the system has an infinite number of solutions when the two lines are the same.

From the provided options:

  • The first option shows a single line.
  • The second option shows two intersecting lines (which indicates a single solution).
  • The third option shows two parallel lines (which indicates no solution).
  • The fourth option shows another single line.

The correct choice, which indicates the lines coincide (and thus there are infinitely many solutions), corresponds to a graph that depicts just one line, but doesn't have a second separate line to compare.

Thus, the answer is the graph that shows a single line, which can be any of the graphs that demonstrates this. However, please clarify which one of these options specifically illustrates a scenario with infinite solutions (same graph overlaid) as they don't seem to show an overlapping line.