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To find the equation that matches the given graph, let's first analyze the graph to extract key information:

1. **Slope**: The graph shows a line that goes through the points \((0, -6)\) and \((8, 4)\).
   - The slope \( m \) is calculated using the formula: 
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{4 - (-6)}{8 - 0} = \frac{4 + 6}{8} = \frac{10}{8} = \frac{5}{4}.
   \]

2. **Y-intercept**: The line crosses the y-axis at \( y = -6 \).

Now, the line's equation in slope-intercept form \( y = mx + b \) is:
   \[
   y = \frac{5}{4}x - 6.
   \]

Let's convert this to a standard linear form \( Ax + By = C \):

   - Multiply everything by 4 to eliminate the fraction:
   \[
   4y = 5x - 24.
   \]
   - Rearrange it to the standard form:
   \[
   5x - 4y = 24.
   \]

Now, let's examine the provided options to find a matching equation:
1. \( \frac{5}{2}x + 4y = 10 \)
2. \( 4x + \frac{5}{2}y = 10 \)
3. \( 8x + 5y = 40 \)
4. \( 5x - 8y = 40 \)

Let's analyze each option to see if any of them match the equation \(5x - 4y = 24\):

- Option 1: Multiply by 2 to clear the fraction:
   \[
   5x + 8y = 20.
   \]
   This is not equivalent to \(5x - 4y = 24\).

- Option 2: Multiply by 2 to clear the fraction:
   \[
   8x + 5y = 20.
   \]
   This does not match.

- Option 3: The equation is \(8x + 5y = 40\). Dividing by 8, we get:
   \[
   x + \frac{5}{8}y = 5,
   \]
   which does not match either.

- Option 4: The equation is \(5x - 8y = 40\), which can be rewritten in slope-intercept form:
   \[
   5x - 40 = 8y,
   \]
   or
   \[
   y = \frac{5}{8}x - 5,
   \]
   which does not match.

None of the options seem to match the derived equation exactly. There might be a simplification or rewriting in another form we're missing. However, based on the given options, the graph best matches **none of these options exactly**. 

Do you want to go into more detail or have any questions?

Here are five related questions for further exploration:
1. How do you convert the slope-intercept form of a line to its standard form?
2. What is the process for finding the slope of a line given two points?
3. How can you identify if a line equation matches a given graph?
4. What changes in an equation if the line is parallel to another?
5. How does the y-intercept affect the position of a line on the graph?

**Tip:** Always double-check the options provided by simplifying or converting them to different forms to find potential matches.

Identify the correct equation that represents the line on a graph using slope-intercept form and convert it into standard form. The line passes through specific points, and the slope is calculated using these coordinates.

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