For a linear system to have an infinite number of solutions, the two equations must represent the same line. This occurs when the equations are equivalent, meaning they have the same slope and y-intercept.
Let's analyze the two equations given:
- Equation 1: \( y = -x + 6 \)
- Equation 2: \( 3x + 3y = 18 \)
We can simplify Equation 2 to solve for \(y\):
\[ 3y = 18 - 3x \ y = -x + 6 \]
Equation 2 is equivalent to Equation 1. Therefore, they represent the same line.
The correct graph that shows these two equations has to be one in which both lines are plotted on top of each other (the same line), indicating that there are infinitely many solutions.
From your descriptions:
- The first option describes two parallel lines, which indicates no solutions.
- The second option describes only one line, which indicates one solution (or infinitely many because it is the same line).
- The third option also describes only one line, similar to the second.
- The fourth option describes two intersecting lines, indicating one solution.
None of the options describe two identical lines, but based on the context in your question, the option indicating one line would have to represent an infinite number of solutions, so it would be either the second or third option.
If you're asked to identify one that represents the infinite number of solutions, you would consider the option depicting the single downward slanting line, as it implies both equations represent the same line.
You can choose either the second or third option based on the description of a single line graph, as they imply infinite solutions. If you have to select just one, choose the third option, as it mentions the line passing through the points (0, 6) and (6, 0) that matches the equations.