To find the equation of the line passing through the point $(3, -5)$ and parallel to the line $3x - 8y = 9$, we need to follow these steps:

1. Find the slope of the given line:

The equation of the line is in standard form:
$3x - 8y = 9$

We need to convert this to slope-intercept form $y = mx + b$ to easily identify the slope $m$. Here's the process:

$3x - 8y = 9$ Subtract $3x$ from both sides: $-8y = -3x + 9$ Now, divide everything by $-8$: $y = \frac{3}{8}x - \frac{9}{8}$

From this, we can see that the slope $m$ of the given line is: $m = \frac{3}{8}$

2. Use the same slope for the parallel line:

Since parallel lines have the same slope, the slope of the new line will also be $\frac{3}{8}$.

3. Write the equation of the new line using point-slope form:

The point-slope form of the equation is: $y - y_1 = m(x - x_1)$ Where $(x_1, y_1)$ is the given point $(3, -5)$, and $m = \frac{3}{8}$.

Substitute the values: $y - (-5) = \frac{3}{8}(x - 3)$ Simplify: $y + 5 = \frac{3}{8}(x - 3)$ Now, distribute $\frac{3}{8}$: $y + 5 = \frac{3}{8}x - \frac{9}{8}$ Subtract 5 from both sides: $y = \frac{3}{8}x - \frac{9}{8} - 5$ Convert 5 to a fraction with denominator 8: $y = \frac{3}{8}x - \frac{9}{8} - \frac{40}{8}$ Now combine the constants: $y = \frac{3}{8}x - \frac{49}{8}$

Final Answer:

The equation of the line containing the point $(3, -5)$ and parallel to $3x - 8y = 9$ is: $y = \frac{3}{8}x - \frac{49}{8}$

Would you like any further details or have any questions?

Related Questions:

1. How would you write the equation of a line parallel to a given line in point-slope form?
2. How can you find the slope of a line from an equation in standard form?
3. How do you convert an equation from standard form to slope-intercept form?
4. What is the difference between parallel and perpendicular lines?
5. How do you write the equation of a line passing through two points?

Tip:

To identify the slope of any line from the equation in standard form $Ax + By = C$, rearrange the equation into slope-intercept form $y = mx + b$, where the slope $m$ is $-A/B$.