To find the tangent of the angle between the line with the equation $3y = 5 - 4x$ and the positive direction of the x-axis, we need to determine the slope of the line. Once we have the slope, we can use it to calculate the tangent of the angle between the line and the x-axis.

Step 1: Find the slope of the line

The given equation is: $3y = 5 - 4x$

First, rewrite the equation in slope-intercept form $y = mx + b$, where $m$ is the slope.

$y = \frac{5 - 4x}{3}$ $y = -\frac{4}{3}x + \frac{5}{3}$

Now, the slope $m$ of the line is $-\frac{4}{3}$.

Step 2: Use the formula for the tangent of the angle

The tangent of the angle $\theta$ between the line and the x-axis is equal to the absolute value of the slope. In this case: $\tan(\theta) = \left| m \right| = \left| -\frac{4}{3} \right| = \frac{4}{3}$

Final Answer:

The tangent of the angle between the line and the positive direction of the x-axis is $\frac{4}{3}$.

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Would you like to explore more details about this, or have any questions? Here are some related questions to expand the concept:

1. How do you derive the angle from the slope of the line?
2. What is the relationship between the slope of a line and the angle it makes with the x-axis?
3. How would the angle change if the slope of the line were positive instead of negative?
4. How do you find the angle between two lines given their equations?
5. What are some practical applications where the angle between a line and the x-axis is important?

Tip: Always ensure the equation is in slope-intercept form ($y = mx + b$) to directly read off the slope, especially when calculating angles.