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

Question

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 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 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.
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Answer 1

To determine which graph indicates that the linear system

-2x + y = 6
and
4x - 2y = -12

has an infinite number of solutions, we must analyze the equations.

First, let's simplify the second equation:

4x - 2y = -12 can be rewritten as 2x - y = -6 (by dividing the entire equation by 2).

Now we can express both equations in slope-intercept form (y = mx + b):

1. From the first equation (-2x + y = 6), we can rearrange it to:

   y = 2x + 6.

2. From the second equation (2x - y = -6), rearranging it gives us:

   -y = -2x - 6 ⇒ y = 2x + 6.

Both equations can be seen as:

y = 2x + 6.

Since both lines have the same slope (2) and the same y-intercept (6), they are essentially the same line. Therefore, the graph that indicates an infinite number of solutions (meaning the lines overlap perfectly) would show only one line or two identical lines.

Among the options you provided, the graph showing two coinciding lines (which may appear as one line) indicates that the system has an infinite number of solutions.

The option with two parallel lines indicates that the system has no solutions, and graphs with intersecting lines indicate a unique solution.

You should choose the graph that shows one line or two overlapping lines. The second option you described, with two parallel lines, would not be your answer, so look for an overlapping representation in any of the remaining options.