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

and 3x−3y=6
has an infinite number of solutions? Graph 1,2,3,4?(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 solid upward slanting line with arrows at both ends passes through the points left parenthesis negative 2 comma 0 right parenthesis and left parenthesis 0 comma 2 right parenthesis. A dotted upward slanting line with arrows at both ends passes through the points left parenthesis 0 comma negative 2 right parenthesis and left parenthesis 2 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. Two parallel lines are plotted on the plane. A solid upward slanting line with arrows at both ends passes through the points left parenthesis negative 2 comma 0 right parenthesis and left parenthesis 0 comma 2 right parenthesis. A dotted upward slanting line with arrows at both ends passes through the points left parenthesis 0 comma negative 2 right parenthesis and left parenthesis 2 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 0 comma negative 2 right parenthesis and left parenthesis 2 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 2 right parenthesis and left parenthesis 2 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 downward slanting line is plotted on the plane. A solid downward slanting line with arrows at both ends passes through left parenthesis negative 2 comma 0 right parenthesis and left parenthesis 0 comma negative 2 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 downward slanting line is plotted on the plane. A solid downward slanting line with arrows at both ends passes through left parenthesis negative 2 comma 0 right parenthesis and left parenthesis 0 comma negative 2 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 6 comma 0 right parenthesis and left parenthesis 5 comma 10 right parenthesis. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis negative 6 comma 4 right parenthesis and left parenthesis 4 comma negative 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. 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 6 comma 0 right parenthesis and left parenthesis 5 comma 10 right parenthesis. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis negative 6 comma 4 right parenthesis and left parenthesis 4 comma negative 6 right parenthesis.
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1 answer

For a linear system to have an infinite number of solutions, the two equations must represent the same line. This means that the lines must be identical, not just parallel or intersecting at a single point.

Given the equations \( y = x - 2 \) and \( 3x - 3y = 6 \):

  1. The first equation, \( y = x - 2 \), is already in slope-intercept form.
  2. The second equation can be rearranged: \[ 3x - 3y = 6 \implies -3y = -3x + 6 \implies y = x - 2 \] Thus, both equations are actually the same line.

For the graphs:

  • You would need a graph showing two identical lines on top of each other, indicating that they are the same line.

From your description:

  • Graph 1: Shows two parallel lines (not the same line).
  • Graph 2: Shows a single line (not two lines).
  • Graph 3: Shows a single downward slanting line (not two lines).
  • Graph 4: Shows two intersecting lines (meaning they have a single solution).

None of the given options indicated a graph with two identical lines. If I were to choose based on this information, I can't positively identify a graph; however, none seem to indicate an infinite number of solutions.

You could respond by looking for any option that may have been missed or report back if there are additional graphs available.