1. The problem involves understanding and analyzing the given equations: - $y = 1 - x$ - $y = x$ - $x - y = 1$ - $x = 1$ - $y = -x - 1$ 2. These are linear equations representing lines in the coordinate plane. 3. Let's rewrite each equation in slope-intercept form $y = mx + b$ where possible: - $y = 1 - x$ can be written as $y = -x + 1$ - $y = x$ is already in slope-intercept form with slope $1$ and intercept $0$ - $x - y = 1$ rearranged gives $y = x - 1$ - $x = 1$ is a vertical line where $x$ is always $1$ - $y = -x - 1$ is already in slope-intercept form with slope $-1$ and intercept $-1$ 4. Each line can be graphed and analyzed for intersections and relative positions. 5. For example, the intersection of $y = x$ and $y = 1 - x$ is found by setting $x = 1 - x$ which gives $2x = 1$ or $x = \frac{1}{2}$, then $y = \frac{1}{2}$. 6. Similarly, intersections and relationships can be found for other pairs. Final answer: The equations represent five distinct lines with slopes and intercepts as identified, useful for graphing and analyzing their intersections.