To find the equation of the line that passes through the point $(4, -4)$ and is parallel to the line given by $3x - 4y = 16$, follow these steps:

Step 1: Find the slope of the given line

The general form of a line is $Ax + By = C$. To find the slope, we first convert the given line equation $3x - 4y = 16$ into the slope-intercept form $y = mx + b$, where $m$ represents the slope.

Starting with: $3x - 4y = 16$

Solve for $y$: $-4y = -3x + 16$ $y = \frac{3}{4}x - 4$

So, the slope of the given line is $m = \frac{3}{4}$.

Step 2: Use the point-slope form

Since the new line is parallel, it will have the same slope $m = \frac{3}{4}$. Now, using the point-slope form of the equation of a line:

$y - y_1 = m(x - x_1)$

where $(x_1, y_1)$ is the given point $(4, -4)$ and $m = \frac{3}{4}$.

Substitute the values: $y - (-4) = \frac{3}{4}(x - 4)$ $y + 4 = \frac{3}{4}(x - 4)$

Step 3: Simplify the equation

Distribute $\frac{3}{4}$: $y + 4 = \frac{3}{4}x - 3$

Subtract 4 from both sides: $y = \frac{3}{4}x - 3 - 4$ $y = \frac{3}{4}x - 7$

Final Answer:

The equation of the line that passes through the point $(4, -4)$ and is parallel to the line $3x - 4y = 16$ is: $y = \frac{3}{4}x - 7$

Would you like a detailed explanation of any of the steps?

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Here are 5 related questions:

1. How do you find the equation of a line given two points?
2. What is the slope of a line perpendicular to a given slope?
3. How do you convert a line from standard form to slope-intercept form?
4. What is the point-slope form of a line, and how is it used?
5. How would the equation change if the line passed through a different point?

Tip: Parallel lines always have the same slope but different $y$-intercepts.