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).
- Divide the entire equation by 3: \[ x + y = 6 \]
- 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:
-
Graph 1 shows two lines, one for \( y = -x + 6 \) and another that does not coincide with it. This does not indicate infinite solutions.
-
Graph 2 shows only the line \( y = -x + 6 \).
-
Graph 3 does not show either line \( y = -x + 6 \).
-
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.
1 answer