Question

consider the line y = -x + 4. what is the slope of a line perpendicular to this line? what is the slope of a line parallel to this line? slope of a perpendicular line: slope of a parallel line:

Explanation:

Step1: Identify the slope of the given line

The equation of the line is given in slope - intercept form \(y = mx + b\), where \(m\) is the slope and \(b\) is the y - intercept. For the line \(y=-x + 4\), we can rewrite it as \(y=(-1)x + 4\). So the slope of the given line \(m=-1\).

Step2: Find the slope of a parallel line

Parallel lines have the same slope. If a line is parallel to the line \(y=-x + 4\), then its slope \(m_{parallel}\) is equal to the slope of the given line. So \(m_{parallel}=-1\).

Step3: Find the slope of a perpendicular line

The product of the slopes of two perpendicular lines (non - vertical and non - horizontal) is \(- 1\). Let the slope of the perpendicular line be \(m_{perpendicular}\). We know that \(m\times m_{perpendicular}=-1\). Since \(m = - 1\), we substitute into the equation: \((-1)\times m_{perpendicular}=-1\). Solving for \(m_{perpendicular}\), we divide both sides of the equation by \(-1\): \(m_{perpendicular}=\frac{-1}{-1}=1\).

Answer:

Slope of a perpendicular line: \(1\)
Slope of a parallel line: \(-1\)