One Solution, No Solution, or Many Solutions Quick Check

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Question
Which of these graphs shows that the linear system y=−x+6 and 3x+3y=18 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. A downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 6 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. A downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 6 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 downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 6 comma 0 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 3 comma 3 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 downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 6 comma 0 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 3 comma 3 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. A downward slanting line with arrows at both ends passes through the points left parenthesis negative 6 comma 0 right parenthesis and left parenthesis 0 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. A line is plotted on the plane. A downward slanting line with arrows at both ends passes through the points left parenthesis negative 6 comma 0 right parenthesis and left parenthesis 0 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 downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 6 comma 0 right parenthesis. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 4 right parenthesis and left parenthesis 4 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 downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 6 right parenthesis and left parenthesis 6 comma 0 right parenthesis. A dotted downward slanting line with arrows at both ends passes through the points left parenthesis 0 comma 4 right parenthesis and left parenthesis 4 comma 0 right parenthesis.
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page 20 of 21

Answer 1

To find which graph shows that the linear system \( y = -x + 6 \) and \( 3x + 3y = 18 \) has an infinite number of solutions, we first need to determine if the two equations represent the same line.

1. Convert the second equation to slope-intercept form:
   Starting from \( 3x + 3y = 18 \): \[ 3y = -3x + 18 \ y = -x + 6 \] This matches the form of the first equation \( y = -x + 6 \).

Since both equations represent the same line, this means they will overlap entirely, resulting in an infinite number of solutions.

2. Finding the correct graph:
   Look for a graph that shows the same line being plotted twice (essentially layered on top of each other).

From your descriptions:

• The graph with two intersecting lines shows distinct lines and is not the correct answer (as intersecting lines represent one solution).
• The graph with two parallel lines also does not represent infinite solutions since parallel lines have no points in common.
• The graphs showing only one line also do not indicate infinitely many solutions since only one line can be present.

The correct choice is therefore the one with two identical lines plotted on top of each other, which represents the same line with the infinite solutions. This could be inferred if such a combination was present (though none is explicitly described above).

If you have the option that depicts the same line multiple times (which may be included in the descriptions you provided), that would be the answer. If not, please review the visual representations again to find one that clearly shows the same line twice.