To solve this problem, we need to determine the slope and $y$-intercept of the line containing $C(0, 0)$ that is perpendicular to line $AB$, and then find the sum of these values.

Step 1: Determine the slope of line $AB$

The coordinates of points $A$ and $B$ are $A(4, 0)$ and $B(0, -3)$. The slope of a line passing through two points is calculated as: $m_{AB} = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting: $m_{AB} = \frac{-3 - 0}{0 - 4} = \frac{-3}{-4} = \frac{3}{4}$

Thus, the slope of $AB$ is $m_{AB} = \frac{3}{4}$.

Step 2: Slope of the perpendicular line

The slope of a line perpendicular to another line is the negative reciprocal of the original slope. For $m_{AB} = \frac{3}{4}$, the perpendicular slope is: $m_{\perp} = -\frac{1}{m_{AB}} = -\frac{4}{3}$

Step 3: Equation of the perpendicular line

The line containing $C(0, 0)$ and perpendicular to $AB$ has slope $m_{\perp} = -\frac{4}{3}$. The equation of a line can be written as: $y = mx + b$ Since the line passes through $C(0, 0)$, the point satisfies the equation, and the $y$-intercept is: $b = 0$ Thus, the equation of the line is: $y = -\frac{4}{3}x$

Step 4: Sum of the slope and $y$-intercept

The slope is $m = -\frac{4}{3}$, and the $y$-intercept is $b = 0$. The sum is: $\text{Sum} = m + b = -\frac{4}{3} + 0 = -\frac{4}{3}$

Final Answer:

The sum of the slope and $y$-intercept for the line is: $\boxed{-\frac{4}{3}}$

Do you want details or have any questions about the solution?

Here are 5 related questions to explore further:

1. What is the general method for finding the equation of a line perpendicular to another line?
2. How would the solution change if $C$ were not at the origin?
3. What are the steps to verify the perpendicularity of two lines algebraically?
4. How do you derive the distance from a point to a line using the line equation?
5. Can we find the point of intersection of two lines if their equations are known?

Tip: When finding slopes, always double-check signs to ensure correct calculations for perpendicular lines.