How do you find the gradient of a line perpendicular to y = 4x - 3?

The gradient of a line perpendicular to y = 4x - 3 is -1/4.

To find the gradient of a line perpendicular to another, you need to understand the relationship between their gradients. The gradient of a line is a measure of its steepness and is represented by the coefficient of \( x \) in the equation \( y = mx + c \), where \( m \) is the gradient. For the line given by \( y = 4x - 3 \), the gradient \( m \) is 4.

When two lines are perpendicular, the product of their gradients is always -1. This means if you multiply the gradient of one line by the gradient of the line perpendicular to it, the result will be -1. Mathematically, if \( m_1 \) is the gradient of the first line and \( m_2 \) is the gradient of the perpendicular line, then \( m_1 \times m_2 = -1 \).

Given that the gradient \( m_1 \) of the line \( y = 4x - 3 \) is 4, we can find the gradient \( m_2 \) of the perpendicular line by solving the equation \( 4 \times m_2 = -1 \). Dividing both sides by 4, we get \( m_2 = -1/4 \).

So, the gradient of a line perpendicular to \( y = 4x - 3 \) is \( -1/4 \). This negative reciprocal relationship ensures that the lines intersect at a right angle, making them perpendicular.