Find the equation of the line passing through the point `(5,2)` and perpendicular to the line joining the points (2,3) and (3,-1).

Explanation

To find the equation of the line passing through the point (5, 2) and perpendicular to the line joining the points (2, 3) and (3, -1), we can follow these steps:

### Step 1: Find the slope of the line joining the points (2, 3) and (3, -1).
The formula for the slope (m) between two points (x1, y1) and (x2, y2) is given by:

\[ m = \frac{y2 - y1}{x2 - x1} \]

Substituting the points (2, 3) and (3, -1):

\[ m_{2} = \frac{-1 - 3}{3 - 2} = \frac{-4}{1} = -4 \]

### Step 2: Determine the slope of the perpendicular line.
If the slope of the first line is \( m_2 \), then the slope of the line perpendicular to it, \( m_1 \), is given by:

\[ m_1 = -\frac{1}{m_2} \]

Substituting \( m_2 = -4 \):

\[ m_1 = -\frac{1}{-4} = \frac{1}{4} \]

### Step 3: Use the point-slope form of the equation of a line.
The point-slope form of the equation of a line is:

\[ y - y_1 = m(x - x_1) \]

Here, \( (x_1, y_1) = (5, 2) \) and \( m = \frac{1}{4} \):

\[ y - 2 = \frac{1}{4}(x - 5) \]

### Step 4: Simplify the equation.
To eliminate the fraction, we can multiply both sides by 4:

\[ 4(y - 2) = x - 5 \]

Expanding this gives:

\[ 4y - 8 = x - 5 \]

### Step 5: Rearrange to standard form.
Rearranging the equation to get all terms on one side:

\[ 4y - x - 8 + 5 = 0 \]

This simplifies to:

\[ 4y - x - 3 = 0 \]

### Final Answer:
The equation of the line is:

\[ 4y - x - 3 = 0 \]

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