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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?(1 point)
Responses

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. 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. 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.

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.

1 answer

To determine which graph shows that the linear system \(y = x - 2\) and \(3x - 3y = 6\) has an infinite number of solutions, we need to check whether the two equations represent the same line.

  1. The equation \(y = x - 2\) is already in slope-intercept form (\(y = mx + b\)), where the slope \(m = 1\) and the y-intercept \(b = -2\).

  2. The second equation \(3x - 3y = 6\) can be rearranged to slope-intercept form. Start by solving for \(y\): \[ 3x - 3y = 6 \ -3y = -3x + 6 \ y = x - 2. \]

Now we see that both equations \(y = x - 2\) and \(y = x - 2\) are indeed the same line. Since they represent the same line, the system has an infinite number of solutions.

Therefore, the correct graph will be the one showing a solid line for \(y = x - 2\) and the same line drawn over it, confirming they are the same line. This would correspond to the fourth graph you described: "Two parallel lines are plotted on the plane," but one of those lines is not actually parallel because it overlaps, indicating that the two equations describe the same line.

The graph you are looking for should show:

  • A line representing \(y = x - 2\)
  • Another line coinciding with it (i.e., the same line).

If you do not see any lines that represent only one unique line, that would indicate an infinite number of solutions. Thus, confirm that a solid downward slanting line is drawn, and you should choose the option that reflects that.