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. 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.

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.

Answer 1

For the linear system given by the equations \( y = -x + 6 \) and \( 3x + 3y = 18 \), we want to determine whether the system has an infinite number of solutions.

First, let's rewrite the second equation in slope-intercept form to understand its relationship to the first equation:

\[
3x + 3y = 18
\]

Dividing the entire equation by 3 gives:

\[
x + y = 6 \\
y = -x + 6
\]

Now we see that both equations are identical since they represent the same line: \( y = -x + 6 \).

When two linear equations represent the same line, they have an infinite number of solutions because any point on that line satisfies both equations.

Now, let’s review the graphs described:

1. **Graph A**: Shows two parallel lines.
2. **Graph B**: Shows two intersecting lines.
3. **Graph C**: Shows a single line.
4. **Graph D**: Shows another single line.

The graph that depicts the infinite number of solutions is one where both equations are represented as the same line, which corresponds to a single line being plotted.

Therefore, **the correct option for a graph that shows the linear system has an infinite number of solutions is one that shows a single line**, which aligns with either **Graph C or Graph D** based on their descriptions. Since we need the graph representing both the equations as one, the graph that provides this is the one depicting the line \( y = -x + 6 \).

If you need to choose from the options listed, it would be the graph depicting the single line \( y = -x + 6 \).