https://file.aiquickdraw.com/mathbot/file/1725527052611bze5r7m2.jpg

To address the question and analyze the graphs in the image, we can break down each graph and derive the corresponding mathematical rule based on the lines shown.

### Graph (a):
- The line passes through points \((0, a)\) on the y-axis and \((b, 0)\) on the x-axis.
- This is a linear equation, and the general form of a line is \( y = mx + c \), where \( m \) is the slope and \( c \) is the y-intercept.
- Slope \(m\) can be calculated as:
  \[
  m = \frac{0 - a}{b - 0} = -\frac{a}{b}
  \]
  So the equation of the line is:
  \[
  y = -\frac{a}{b}x + a
  \]

### Graph (b):
- The line passes through points \((0, -a)\) on the y-axis and \((b, 0)\) on the x-axis.
- The slope of the line:
  \[
  m = \frac{0 - (-a)}{b - 0} = \frac{a}{b}
  \]
  The equation of the line becomes:
  \[
  y = \frac{a}{b}x - a
  \]

### Graph (c):
- The line passes through points \((0, a)\) on the y-axis and \((b, 0)\) on the x-axis.
- Here, the slope:
  \[
  m = \frac{0 - a}{b - 0} = -\frac{a}{b}
  \]
  The equation is similar to graph (a):
  \[
  y = -\frac{a}{b}x + a
  \]

### Summary of equations:
- **(a)**: \( y = -\frac{a}{b}x + a \)
- **(b)**: \( y = \frac{a}{b}x - a \)
- **(c)**: \( y = -\frac{a}{b}x + a \)

Would you like further details or have any questions?

Here are some related questions:
1. How do you calculate the slope of a line from two points?
2. How does changing the sign of the slope affect the orientation of a line?
3. Can you express the equation of a line in point-slope form?
4. How can you determine the intercepts of a line given its equation?
5. What is the geometric significance of the slope \( \frac{a}{b} \) in these cases?

**Tip**: To quickly find the slope of a line, divide the vertical change by the horizontal change between any two points on the line.

Write the rule for the given graphs. Your rule could include pronumerals x and/or b.

Explore how to interpret graphs and write the equations for lines using the pronumerals x and b. This involves understanding linear equations, calculating slope, and deriving the equation in slope-intercept form. Suitable for students in Grades 9-10.

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