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. 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.

B. 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.

C. 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.

D. 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

just do one thats closest

Answer 2

To determine which graph represents the system of equations having an infinite number of solutions, we need to check if the two equations represent the same line.

The equations given are:
1. \( y = -x + 6 \)
2. \( 3x + 3y = 18 \)

We can rewrite the second equation in slope-intercept form to see if it's equivalent to the first equation:

1. Start with \( 3x + 3y = 18 \).
2. Divide everything by 3:
\[ x + y = 6 \]
3. Rearranging gives:
\[ y = -x + 6 \]

Now we can see that both equations are the same, \( y = -x + 6 \). This means that the lines represented by both equations coincide, which indicates that there are infinitely many solutions.

Now, let's analyze the choices:

- **Choice A**: Two intersecting lines (not the same line, so not infinite solutions).
- **Choice B**: One line only (not infinite solutions).
- **Choice C**: Two parallel lines (not the same line, so not infinite solutions).
- **Choice D**: A single line (not infinite solutions).

None of the choices provided represents the situation where the two lines overlap completely. Therefore, none of the graphs show that the linear system \( y = -x + 6 \) and \( 3x + 3y = 18 \) has an infinite number of solutions.

If you have the option to present another graph that shows both lines coinciding, that would represent an infinite number of solutions.

Answer 3

Given the options and the need to choose the closest representation of the system of equations \( y = -x + 6 \) and \( 3x + 3y = 18 \) (which we showed are equivalent), the most appropriate choice from the provided options is:

**Choice D**: A downward slanting line that passes through the points \( (0, 6) \) and \( (6, 0) \).

While it doesn't show both equations (as they are the same line), it at least depicts the correct line whose equation represents both \( y = -x + 6 \) and \( 3x + 3y = 18 \).

So, the answer would be **D**.