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

The equation of a line perpendicular to \( y = x - 5 \) is \( y = -x + c \), where \( c \) is a constant.

To understand why, let's start by looking at the gradient (or slope) of the given line \( y = x - 5 \). The gradient of this line is 1, as it is the coefficient of \( x \). For two lines to be perpendicular, the product of their gradients must be -1. This means if one line has a gradient of \( m \), the perpendicular line must have a gradient of \( -\frac{1}{m} \).

In this case, the gradient of the given line is 1. Therefore, the gradient of the perpendicular line must be \( -\frac{1}{1} = -1 \). So, the gradient of the perpendicular line is -1.

Now, we can write the equation of the perpendicular line. The general form of a straight line equation is \( y = mx + c \), where \( m \) is the gradient and \( c \) is the y-intercept. Since the gradient of our perpendicular line is -1, the equation becomes \( y = -x + c \), where \( c \) is a constant that can be any real number.

This constant \( c \) determines where the line crosses the y-axis. For example, if \( c = 2 \), the equation of the line would be \( y = -x + 2 \). If \( c = -3 \), the equation would be \( y = -x - 3 \). The value of \( c \) can vary depending on the specific line you are looking for, but the gradient will always be -1 for a line perpendicular to \( y = x - 5 \).