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

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 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. 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 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.

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.
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GPT-4o mini · Answer

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

The first equation, \( y = -x + 6 \), can be rewritten in standard form as \( x + y = 6 \).
For the second equation, \( 3x + 3y = 18 \), we can simplify it by dividing everything by 3, which gives us \( x + y = 6 \).

Since both equations simplify to the same line, the linear system has an infinite number of solutions.

Now, looking at the provided graph descriptions:

First graph: A downward slanting line through points (-6, 0) and (0, -6); this does not represent \( y = -x + 6 \).
Second graph: A downward slanting line through points (0, 6) and (6, 0); this represents \( y = -x + 6 \).
Third graph: Two parallel lines; this represents two different equations that do not intersect, thus they do not yield an infinite number of solutions.
Fourth graph: Two intersecting lines; this represents two different equations that intersect at one point, thus they yield a single solution.

The only graph that shows one line representing both equations is the second graph. Therefore, the correct response is:

A downward slanting line through points (0, 6) and (6, 0) representing the equation \( y = -x + 6 \), confirming that the system has an infinite number of solutions since the other equation simplifies to that.