What is the gradient of a line perpendicular to y = 5x - 2?

The gradient of a line perpendicular to \( y = 5x - 2 \) is \( -\frac{1}{5} \).

To understand why, let's start by looking at the given equation \( y = 5x - 2 \). This is in the form \( y = mx + c \), where \( m \) represents the gradient of the line. In this case, the gradient \( m \) is 5.

When two lines are perpendicular, the product of their gradients is always -1. This is a key property in coordinate geometry. So, if the gradient of one line is \( m \), the gradient of the line perpendicular to it will be \( -\frac{1}{m} \).

Given that the gradient of the original line is 5, we can find the gradient of the perpendicular line by taking the negative reciprocal of 5. The negative reciprocal of 5 is \( -\frac{1}{5} \). Therefore, the gradient of the line perpendicular to \( y = 5x - 2 \) is \( -\frac{1}{5} \).

This concept is very useful in various problems involving perpendicular lines, such as finding the equation of a line that is perpendicular to a given line and passes through a specific point. Remember, the key step is to take the negative reciprocal of the given gradient.