To find the equation of line $Q$, we need to determine the slope $m$ and the y-intercept $c$ in the form $y = mx + c$.

Step 1: Find the Slope

Since line $Q$ is parallel to the line given by $3y = 12x - 5$, it will have the same slope as this line. Let's rewrite the equation $3y = 12x - 5$ in slope-intercept form (i.e., $y = mx + c$).

1. Divide both sides by 3: $y = 4x - \frac{5}{3}$

So, the slope $m$ of this line is $4$. Since line $Q$ is parallel to this line, it also has a slope of $4$.

Step 2: Use the Point-Slope Form

Now we know that the slope of line $Q$ is $4$, and it passes through the point $(-2, -11)$. We can use the point-slope form of a line: $y - y_1 = m(x - x_1)$ where $(x_1, y_1) = (-2, -11)$ and $m = 4$.

Substitute these values: $y - (-11) = 4(x - (-2))$ $y + 11 = 4(x + 2)$

Step 3: Simplify to Slope-Intercept Form

Expand and simplify: $y + 11 = 4x + 8$ $y = 4x + 8 - 11$ $y = 4x - 3$

Final Answer

The equation of line $Q$ is: $y = 4x - 3$

Let me know if you would like more details on any of the steps or have questions!

Here are some related questions:

1. How do you find the equation of a line perpendicular to a given line?
2. How would the equation change if line $Q$ passed through a different point?
3. How can you find the y-intercept directly if given two points?
4. What are some methods to verify that two lines are parallel?
5. How can you use this information to find equations of lines in three-dimensional space?

Tip: Remember, parallel lines have equal slopes, and perpendicular lines have slopes that are negative reciprocals of each other.