To determine if the line given by the equation $2x + 3y + 4 = 0$ is parallel to the line passing through the points $(9, 4)$ and $(3, 8)$, follow these steps:

Step 1: Find the slope of the given line $2x + 3y + 4 = 0$.

To find the slope of the line, we rewrite the equation in the slope-intercept form $y = mx + b$, where $m$ is the slope.

$2x + 3y + 4 = 0$ Solve for $y$: $3y = -2x - 4$ $y = -\frac{2}{3}x - \frac{4}{3}$ Thus, the slope of the line is $m = -\frac{2}{3}$.

Step 2: Find the slope of the line passing through the points $(9, 4)$ and $(3, 8)$.

The slope $m$ of a line passing through two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by the formula: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substituting the points $(9, 4)$ and $(3, 8)$: $m = \frac{8 - 4}{3 - 9} = \frac{4}{-6} = -\frac{2}{3}$

Step 3: Compare the slopes.

The slope of the given line is $m = -\frac{2}{3}$, and the slope of the line passing through the points $(9, 4)$ and $(3, 8)$ is also $m = -\frac{2}{3}$.

Since the slopes are equal, the two lines are parallel.

Conclusion:

The line given by the equation $2x + 3y + 4 = 0$ is parallel to the line passing through the points $(9, 4)$ and $(3, 8)$.

Would you like further details on any part of the solution? Here are some related questions to think about:

1. What is the slope-intercept form of a line and how is it derived?
2. How do you find the equation of a line given two points?
3. What conditions must be satisfied for two lines to be perpendicular?
4. How can you find the distance between a point and a line?
5. What happens if two lines have the same slope but different intercepts?

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