What is the gradient of a line that is perpendicular to the line represented by $y=\frac{1}{3}x+5$?

The correct answer is -3.

To find the gradient (slope) of a line that is perpendicular to a given line, we can follow these steps:

Step 1: Identify the gradient of the given line.

The standard equation of a straight line in slope-intercept form is:

$y=mx+c$

where $m$ represents the gradient of the line and $c$ represents the y-intercept.

The given line equation is:

$y=\frac{1}{3}x+5$

Comparing this with the standard slope-intercept form, we see that the gradient of the given line is:

$m_1=\frac{1}{3}$

Step 2: Calculate the gradient of the perpendicular line.

Two lines are perpendicular if and only if the product of their gradients equals -1 (i.e., they are negative reciprocals of each other):

$m_1\times m_2=-1$

where $m_1$ is the gradient of the first line and $m_2$ is the gradient of the perpendicular line.

Substitute $m_1=\frac{1}{3}$ into the equation:

$\frac{1}{3}\times m_2=-1$

Multiply both sides by 3 to solve for $m_2$:

$m_2=-1\times 3$

$m_2=-3$

Therefore, the gradient of the perpendicular line is -3.