Find the slope of a line perpendicular to the line which passes through each pair of the following points: \[{\text{(}}{{\text{x}}_1}{\text{,}}{{\text{y}}_1}{\text{) and (}}{{\text{x}}_2}{\text{,}}{{\text{y}}_2}{\text{)}}{\text{.}}\]

Answer

To find the slope of a line that is perpendicular to the line passing through the points \((x_1, y_1)\) and \((x_2, y_2)\), we can follow these steps:

### Step 1: Calculate the slope of the line through the given points
The slope \(m_1\) of the line passing through the points \((x_1, y_1)\) and \((x_2, y_2)\) is given by the formula:
\[ m_1 = \frac{y_2 - y_1}{x_2 - x_1} \]

### Step 2: Use the property of perpendicular lines
For two lines to be perpendicular, the product of their slopes must equal \(-1\). If \(m_2\) is the slope of the line perpendicular to the line with slope \(m_1\), then:
\[ m_1 \cdot m_2 = -1 \]

### Step 3: Solve for the slope of the perpendicular line
From the equation \(m_1 \cdot m_2 = -1\), we can express \(m_2\) in terms of \(m_1\):
\[ m_2 = -\frac{1}{m_1} \]
Substituting the expression for \(m_1\) from Step 1:
\[ m_2 = -\frac{1}{\frac{y_2 - y_1}{x_2 - x_1}} = -\frac{x_2 - x_1}{y_2 - y_1} \]

### Final Result
Thus, the slope of the line that is perpendicular to the line passing through the points \((x_1, y_1)\) and \((x_2, y_2)\) is:
\[ m_2 = \frac{x_1 - x_2}{y_2 - y_1} \]