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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. 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.
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. 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.

1 answer

To determine which graph shows that the linear system given has an infinite number of solutions, we need to recognize that this occurs when the two equations represent the same line. For two equations to be the same line, they must be parallel and coincide exactly, which means they have the same slope and y-intercept.

Now, let’s analyze the equations:

  1. Equation 1: \(-2x + y = 6\)

    • Rearranging it to slope-intercept form (y = mx + b): \[ y = 2x + 6 \]
    • The slope is \(2\) and the y-intercept is \(6\).

  2. Equation 2: \(4x - 2y = -12\)

    • Rearranging it: \[ -2y = -4x - 12 \quad \Rightarrow \quad 2y = 4x + 12 \quad \Rightarrow \quad y = 2x + 6 \]
    • Again, the slope is \(2\) and the y-intercept is \(6\).

Since both equations simplify to the same equation \(y = 2x + 6\), this means that they represent the same line and therefore the system has an infinite number of solutions, as every point on the line is a solution.

Looking at the responses:

  • The first option describes two parallel lines, which indicates the system has no solutions.
  • The second option describes intersecting lines, which indicates a unique solution.
  • The third option describes only one line, which would indicate that it has infinitely many solutions.
  • The fourth option also describes only one line, which would indicate that it has infinitely many solutions.

Therefore, the correct answer that shows the linear system has an infinite number of solutions is the third option, where only one line is plotted on the graph.