Determine the gradient of a line that is perpendicular to the line represented by the equation 5x - 3y + 15 = 0.

The correct answer is:

$-\frac{3}{5}$

To find the gradient of a line that is perpendicular to a given line, we first determine the gradient of the original line.

Step 1: Find the gradient of the given line.

The equation of the given line is:

$5x-3y+15=0$

Rearrange the equation into slope-intercept form, y = mx + c, where m represents the gradient of the line:

$3y=5x+15$

Divide all terms by 3:

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

From this equation, the gradient of the given line, m₁, is:

$m_1=\frac{5}{3}$

Step 2: Find the gradient of the perpendicular line.

Two non-vertical lines are perpendicular if and only if the product of their gradients is equal to -1:

$m_1\times m_2=-1$

Substitute the value of m₁ into the perpendicularity formula:

$\left(\frac{5}{3}\right)\times m_2=-1$

Solve for m₂ by taking the negative reciprocal:

$m_2=-1\times \frac{3}{5}$

$m_2=-\frac{3}{5}$

Therefore, the gradient of the perpendicular line is:

$-\frac{3}{5}$