Let's solve the problem presented in your image step by step:

The equation of the line given is:

$3x - 3y = 63$

Step 1: Rewrite the equation in slope-intercept form (y = mx + b)

First, we solve for $y$.

$3x - 3y = 63$

Divide the entire equation by 3:

$x - y = 21$

Now, solve for $y$:

$y = x - 21$

So, the equation of the line is now in the form $y = mx + b$, where $m = 1$. This means the slope of the original line is 1.

Step 2: Find the slope of the perpendicular line

The slopes of perpendicular lines are negative reciprocals of each other. The slope of the given line is 1, so the slope of the line perpendicular to it will be:

$m_{\perp} = -\frac{1}{1} = -1$

Final Answer:

The slope of the line perpendicular to the given line is -1.

Do you need further clarification, or would you like more details on any part?

Here are 5 related questions to further your understanding:

1. How do you find the slope of a line given its equation in different forms?
2. What is the relationship between the slopes of parallel lines?
3. Can you explain how to convert a line from standard form to slope-intercept form?
4. What is the negative reciprocal of a fraction, and how does it relate to perpendicular lines?
5. How can the y-intercept affect the graph of a line?

Tip: When finding the slope of a line, always aim to rearrange the equation into the form $y = mx + b$, where $m$ is the slope.