How do you find the perpendicular line to y = 5x + 3 through (2, 4)?

To find the perpendicular line to \( y = 5x + 3 \) through \( (2, 4) \), use the negative reciprocal slope.

The slope (or gradient) of the given line \( y = 5x + 3 \) is 5. For a line to be perpendicular to this, its slope must be the negative reciprocal of 5. The negative reciprocal of 5 is \(-\frac{1}{5}\). This means the slope of the perpendicular line is \(-\frac{1}{5}\).

Next, we use the point-slope form of a line equation, which is \( y - y_1 = m(x - x_1) \), where \( m \) is the slope and \( (x_1, y_1) \) is a point on the line. Here, \( m = -\frac{1}{5} \) and the point is \( (2, 4) \).

Substitute the values into the point-slope form:
\[ y - 4 = -\frac{1}{5}(x - 2) \]

Now, simplify this equation to get it into the slope-intercept form \( y = mx + c \):
\[ y - 4 = -\frac{1}{5}x + \frac{2}{5} \]
\[ y = -\frac{1}{5}x + \frac{2}{5} + 4 \]
\[ y = -\frac{1}{5}x + \frac{2}{5} + \frac{20}{5} \]
\[ y = -\frac{1}{5}x + \frac{22}{5} \]

So, the equation of the line perpendicular to \( y = 5x + 3 \) and passing through \( (2, 4) \) is \( y = -\frac{1}{5}x + \frac{22}{5} \).