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When studying lines in geometry, a key concept is determining the relationship between two lines: are they parallel, perpendicular, or neither? Let's start by understanding **paralleling lines**. Parallel lines have identical slopes. This means that no matter how far these lines extend, they will never intersect.
For example, if one line has a slope of 3, another parallel line will also have a slope of 3.
On the other hand, **perpendicular lines** are lines that intersect at a right angle (90 degrees). For two lines to be perpendicular, the product of their slopes must equal -1. This means if one line has a slope of 2, the perpendicular line should have a slope of ewline\(-\frac{1}{2}\).In the given steps, neither parallelism nor perpendicularity is observed because the slopes of the lines are \(-2\) and \(-1\) and their product is 2, not -1. Therefore, the lines are neither parallel nor perpendicular.

The intersection of two lines represents the exact point where the two lines cross each other. This is an important concept in algebra and can be seen in problems involving finding specific coordinates that satisfy both line equations simultaneously.
To find an intersection, we set the equations of the two lines equal to each other and solve for the variable, typically represented by \( x \).
In our example, we take the equations for Line 1, \(f(x) = -2x - 1\), and Line 2, \(g(x) = -x\),. Setting these equal, \(-2x - 1 = -x\), we solve for \( x \), which gives us \( x = -1 \).After finding \( x \), we substitute this value back into one of the original line equations to determine the \( y \)-coordinate, resulting in the intersection point. In this case, substituting into \(g(x) = -x\), we find that \( y = 1 \).Thus, the lines intersect at the point \((-1, 1)\), meaning both lines cross at this specific coordinate on a graph.

Understanding the slope-intercept form of a line is essential for analyzing and graphing linear equations. The general form of a line in this category is given by the equation \( y = mx + b \). Here, \( m \) represents the slope of the line, and \( b \) is the y-intercept, indicating where the line crosses the y-axis.
The **slope** \( m \) tells us how steep the line is. A positive slope means the line rises as it moves from left to right, while a negative slope indicates the line descends.
The steeper the slope, the more inclined the line.Meanwhile, the **y-intercept** \( b \) provides an easy starting point for graphing the line as it specifies where the line begins on the y-axis. Simply plot \((0, b)\) on your graph as a starting place.In the provided solutions, both lines, \(f(x) = -2x - 1\) and \(g(x) = -x\), are in slope-intercept form:

• The slope for \(f(x)\) is \(-2\) and it crosses the y-axis at \(-1\).
• The slope for \(g(x)\) is \(-1\) with a y-intercept of \(0\).

By recognizing these parts of the equation, we can quickly understand and graph the lines, analyzing their direction and way they interact on a Cartesian plane.