To determine the equation of the red line in the form $ax + by = c$, where $a, b, c$ are integers and $a$ is positive, follow these steps:

1. Identify two points on the line:

From the graph:

• One point is $(-2, 4)$.
• Another point is $(4, 0)$.

2. Calculate the slope ($m$):

The slope formula is: $m = \frac{y_2 - y_1}{x_2 - x_1}$ Substitute: $m = \frac{0 - 4}{4 - (-2)} = \frac{-4}{6} = -\frac{2}{3}.$ Thus, the slope is $m = -\frac{2}{3}$.

3. Use the point-slope form of the line equation:

The point-slope form is: $y - y_1 = m(x - x_1).$ Choose the point $(-2, 4)$: $y - 4 = -\frac{2}{3}(x - (-2)).$ Simplify: $y - 4 = -\frac{2}{3}(x + 2),$ $y - 4 = -\frac{2}{3}x - \frac{4}{3}.$ $y = -\frac{2}{3}x - \frac{4}{3} + 4.$ Convert $4$ to $\frac{12}{3}$: $y = -\frac{2}{3}x + \frac{8}{3}.$

4. Rewrite in standard form $ax + by = c$:

Multiply through by 3 to eliminate fractions: $3y = -2x + 8.$ Rearrange: $2x + 3y = 8.$

Final Answer:

The equation of the line is: $2x + 3y = 8.$

Let me know if you'd like additional details or clarifications!

Here are 5 related questions for further exploration:

1. How can you verify if a given point lies on a specific line?
2. What is the difference between slope-intercept form and standard form of a line equation?
3. How would the equation change if the line passed through the origin?
4. What does the slope of the line represent geometrically?
5. Can the coefficients $a, b, c$ of the line equation be negative?

Tip: Always double-check the slope by picking two distinct points on the line to ensure accuracy.