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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.

Determine whether the line with equation 2x + 3y + 4 = 0 is parallel to the line through the points with coordinates (9, 4) and (3, 8).

Learn how to determine if the line with equation 2x + 3y + 4 = 0 is parallel to the line passing through the points (9, 4) and (3, 8). This problem involves calculating slopes and comparing them to check for parallelism, suitable for students in Grades 8-10.

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