Given: The equation of lines as-
[tex]\begin{gathered} y=x-4 \\ y=x+8 \\ y=-x+2 \end{gathered}[/tex]
Required: To identify any relationships that exist among the lines.
Explanation: The slope-intercept form of equation of a line is-
[tex]y=mx+c[/tex]
Hence, comparing the given equations of line with slope intercept form, we get the slopes of the lines as-
[tex]\begin{gathered} m_1=1 \\ m_2=1 \\ m_3=-1 \end{gathered}[/tex]
Now, two lines are parallel if their slopes are equal, and are perpendicular if the product of slopes is -1.
Hence, line (a) y=x-4 is parallel to line (b) y=x+8.
Both these lines are also perpendicular to the line (c) y=-x+2.
The graph of these lines is shown below-
Final Answer: The correct options are- (a) is perpendicular to (c), (a) is parallel to (b), and (b) is perpendicular to (c).
