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The problem asks for the equation of a line in the form \( Ax + By = C \), where the line passes through the point \( (-4, -7) \) and has a slope of \( \frac{5}{2} \).

### Step-by-Step Solution:

1. **Point-Slope Form**:  
   We can start by using the point-slope form of a line:
   \[
   y - y_1 = m(x - x_1)
   \]
   where:
   - \( m = \frac{5}{2} \) (the slope),
   - \( (x_1, y_1) = (-4, -7) \) (the given point).

   Substituting the values:
   \[
   y - (-7) = \frac{5}{2}(x - (-4))
   \]
   Simplifying this:
   \[
   y + 7 = \frac{5}{2}(x + 4)
   \]

2. **Distribute the Slope**:
   Next, distribute \( \frac{5}{2} \) on the right-hand side:
   \[
   y + 7 = \frac{5}{2}x + \frac{5}{2} \times 4
   \]
   \[
   y + 7 = \frac{5}{2}x + 10
   \]

3. **Simplify the Equation**:
   Subtract 7 from both sides to isolate \( y \):
   \[
   y = \frac{5}{2}x + 3
   \]

4. **Convert to Standard Form \( Ax + By = C \)**:
   Multiply the entire equation by 2 to eliminate the fraction:
   \[
   2y = 5x + 6
   \]
   Rearranging terms to put the equation in the form \( Ax + By = C \):
   \[
   5x - 2y = -6
   \]

Thus, the equation of the line is:
\[
5x - 2y = -6
\]

Would you like further details on any step?

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Here are five related questions to deepen your understanding:
1. How do we derive the point-slope form of a line equation?
2. What other forms can we use to express a linear equation?
3. Can we find the x- and y-intercepts from the standard form?
4. How would the equation change if the slope or point were different?
5. How can we graph the line using the equation \( 5x - 2y = -6 \)?

**Tip**: Always simplify the fractions or clear them by multiplying to avoid complications in standard form.

Una recta atraviesa al punto (-4, -7) y tiene una pendiente igual a 5/2. Escribir una ecuación en la forma Ax + By = C para esta recta. Utilizar números enteros para A, B y C.

Learn how to derive the standard form Ax + By = C of a linear equation from a point (-4, -7) with a slope of 5/2, using integer coefficients. This example illustrates the application of point-slope and standard form transformations suitable for students in Grades 8-10.

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