Answer :
To determine which line proves Ellen's statement incorrect, we need to understand what it means for a line to have "no slope." When a line has no slope, it implies that the line is horizontal. Horizontal lines have the equation [tex]\( y = c \)[/tex], where [tex]\( c \)[/tex] is a constant.
Here are the options:
1. [tex]\( x = 0 \)[/tex]: This represents the y-axis itself, a vertical line passing through the origin (0,0). This line is vertical, not horizontal, and thus does not fit the criteria of having no slope.
2. [tex]\( y = 0 \)[/tex]: This represents the x-axis, a horizontal line passing through the origin (0,0). This line definitely has no slope, as it is perfectly horizontal. Moreover, it touches the y-axis at the origin. This fits the criteria of disproving Ellen's statement.
3. [tex]\( x = 1 \)[/tex]: This is a vertical line that crosses the x-axis at [tex]\( x = 1 \)[/tex]. It is vertical and thus does not fit the criteria of having no slope.
4. [tex]\( y = 1 \)[/tex]: This represents a horizontal line that crosses the y-axis at [tex]\( y = 1 \)[/tex]. This line also has no slope and touches the y-axis at [tex]\( y = 1 \)[/tex].
Given the requirement that the line has no slope and touches the y-axis, the correct line is:
[tex]\[ y = 0 \][/tex]
This line is horizontal and touches the y-axis at the origin (0,0), thereby disproving Ellen's statement that a line with no slope never touches the y-axis.
Thus, the line that proves Ellen's statement incorrect is:
[tex]\[ \boxed{y = 0} \][/tex]
Here are the options:
1. [tex]\( x = 0 \)[/tex]: This represents the y-axis itself, a vertical line passing through the origin (0,0). This line is vertical, not horizontal, and thus does not fit the criteria of having no slope.
2. [tex]\( y = 0 \)[/tex]: This represents the x-axis, a horizontal line passing through the origin (0,0). This line definitely has no slope, as it is perfectly horizontal. Moreover, it touches the y-axis at the origin. This fits the criteria of disproving Ellen's statement.
3. [tex]\( x = 1 \)[/tex]: This is a vertical line that crosses the x-axis at [tex]\( x = 1 \)[/tex]. It is vertical and thus does not fit the criteria of having no slope.
4. [tex]\( y = 1 \)[/tex]: This represents a horizontal line that crosses the y-axis at [tex]\( y = 1 \)[/tex]. This line also has no slope and touches the y-axis at [tex]\( y = 1 \)[/tex].
Given the requirement that the line has no slope and touches the y-axis, the correct line is:
[tex]\[ y = 0 \][/tex]
This line is horizontal and touches the y-axis at the origin (0,0), thereby disproving Ellen's statement that a line with no slope never touches the y-axis.
Thus, the line that proves Ellen's statement incorrect is:
[tex]\[ \boxed{y = 0} \][/tex]
