To find the equation of a line that is parallel to a given line and passes through a specific point, follow these steps:

Step 1: Find the slope of the given line

The given line equation is $3x + 4y = 7$.

First, we need to write this equation in the slope-intercept form $y = mx + b$ to find its slope $m$.

Isolate $y$ by solving for it: $4y = -3x + 7$ Divide every term by 4: $y = -\frac{3}{4}x + \frac{7}{4}$ So, the slope $m_1$ of the given line is $-\frac{3}{4}$.

Step 2: The slope of the parallel line

Since parallel lines have the same slope, the slope $m_2$ of the line we need to find is also $-\frac{3}{4}$.

Step 3: Use the point-slope form to write the equation

Now, use the point-slope form of the equation $y - y_1 = m(x - x_1)$, where $(x_1, y_1)$ is the point the line passes through and $m$ is the slope.

The point given is $(-4, -9)$, so: $y - (-9) = -\frac{3}{4}(x - (-4))$ $y + 9 = -\frac{3}{4}(x + 4)$ Distribute $-\frac{3}{4}$ on the right side: $y + 9 = -\frac{3}{4}x - 3$ Finally, subtract 9 from both sides to isolate $y$: $y = -\frac{3}{4}x - 12$

Step 4: Write the equation using function notation

The equation in function notation is: $f(x) = -\frac{3}{4}x - 12$

So, the equation of the line is $f(x) = -\frac{3}{4}x - 12$.

Would you like more details, or do you have any questions?

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1. How do you identify the slope of a line from its standard form?
2. What are the key properties of parallel lines?
3. How do you apply the point-slope form to find a line equation?
4. How do you convert an equation from standard form to slope-intercept form?
5. What is the significance of the slope in determining parallel lines?
6. Can you explain the function notation and its use?
7. How do you check if two lines are parallel?
8. How do you isolate a variable in an algebraic equation?

Tip: For problems involving parallel lines, remember that their slopes are identical, making it easy to find the equation of a parallel line if the slope is known.