What is the equation of a line perpendicular to y = 3x - 1?

The equation of a line perpendicular to \( y = 3x - 1 \) is \( y = -\frac{1}{3}x + c \).

To understand why, let's start by looking at the slope (or gradient) of the given line. The equation \( y = 3x - 1 \) is in the slope-intercept form \( y = mx + c \), where \( m \) represents the slope. Here, the slope \( m \) is 3.

When two lines are perpendicular, their slopes are negative reciprocals of each other. This means if one line has a slope of \( m \), the perpendicular line will have a slope of \( -\frac{1}{m} \). For our given line with a slope of 3, the perpendicular slope will be \( -\frac{1}{3} \).

Now, to write the equation of a line with this new slope, we use the slope-intercept form again: \( y = mx + c \). Substituting the perpendicular slope, we get \( y = -\frac{1}{3}x + c \), where \( c \) is the y-intercept. The value of \( c \) can be any real number, depending on where the line crosses the y-axis.

For example, if the line crosses the y-axis at 2, the equation would be \( y = -\frac{1}{3}x + 2 \). If it crosses at -4, the equation would be \( y = -\frac{1}{3}x - 4 \). The key point is that the slope must be \( -\frac{1}{3} \) for the line to be perpendicular to the original line \( y = 3x - 1 \).