Question

Which of these graphs shows that the linear system y=-x+6 and 3x + 3y = 18 has an infinite number of solutions?

Graph 1 (0,6) (6,0) line 2 (0,4) (4,0)
Graph 2 (0,6) (6,0)
Graph 3 (0,-6) (-6,0)
Graph 4 (0,6) (6,0) line 2 (0,3.25)

Answer 1

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

First, let's rewrite the second equation \( 3x + 3y = 18 \) in slope-intercept form (y = mx + b).

1. Divide the entire equation by 3:
\[
x + y = 6
\]
2. Rearranging gives:
\[
y = -x + 6
\]

Now we can see that both equations are the same:
- \( y = -x + 6 \) (from the first equation)
- \( y = -x + 6 \) (from the second equation after simplification)

This confirms that the two equations represent the same line, meaning there are an infinite number of solutions (every point on the line is a solution).

Next, we analyze the options:

1. **Graph 1** shows two lines, one for \( y = -x + 6 \) and another that does not coincide with it. This does not indicate infinite solutions.

2. **Graph 2** shows only the line \( y = -x + 6 \).

3. **Graph 3** does not show either line \( y = -x + 6 \).

4. **Graph 4** shows the line \( y = -x + 6 \) with another presumably different line that does not match.

The correct answer is **Graph 2**, as it shows the line \( y = -x + 6 \) alone, indicating it is the only one, thus representing an infinite number of solutions.