To find the equation of the red line in standard form, we need to determine two points on the line and use them to calculate the slope and y-intercept.

From the image, two clear points on the line are:

1. $(2, -3)$
2. $(-2, 1)$

Let's find the slope $m$ of the line:

$m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{1 - (-3)}{-2 - 2} = \frac{1 + 3}{-2 - 2} = \frac{4}{-4} = -1$

So, the slope $m$ is $-1$.

Using the point-slope form $y - y_1 = m(x - x_1)$ with the point $(2, -3)$:

$y - (-3) = -1(x - 2)$ $y + 3 = -1(x - 2)$ $y + 3 = -x + 2$ $y = -x + 2 - 3$ $y = -x - 1$

Now, convert this equation to standard form $Ax + By = C$:

$y = -x - 1$ $x + y = -1$

Here, $A = 1$, $B = 1$, and $C = -1$. The greatest common divisor of $1, 1,$ and $-1$ is 1, and $A$ is already positive.

Thus, the standard form of the equation is:

$x + y = -1$

Would you like more details on any part of this process, or do you have any questions? Here are some related questions to consider:

1. How do you determine the slope of a line given two points?
2. What is the point-slope form of a linear equation?
3. How can you convert the slope-intercept form to standard form?
4. What are the properties of the standard form of a linear equation?
5. How do you find the x-intercept and y-intercept of a line from its equation?
6. What is the process for verifying if a point lies on a given line?
7. How can the distance between two points on a line be calculated?
8. What are the applications of linear equations in real-life scenarios?

Tip: Always ensure your final equation in standard form has integer coefficients and the coefficient of $x$ is positive.