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To solve this problem, we need to determine the equations of the lines labeled A, B, C, D, and E based on the graph provided.

### Step-by-Step Process:

1. **Identify two points on each line**: From the graph, pick two points for each labeled line that are easy to read, like points where the line crosses grid intersections.

2. **Use the slope formula**:  
   \[
   m = \frac{y_2 - y_1}{x_2 - x_1}
   \]
   The slope \( m \) describes how steep the line is.

3. **Determine the y-intercept**: This is the point where the line crosses the \( y \)-axis (where \( x = 0 \)).

4. **Form the equation**: The equation of a line can be written in the form:
   \[
   y = mx + b
   \]
   Where \( m \) is the slope and \( b \) is the y-intercept.

Let’s begin by analyzing the points on the graph for each line:

### Line A:
- Points: \( (-7, 0) \) and \( (0, 2) \)
- Slope \( m_A \):
  \[
  m_A = \frac{2 - 0}{0 - (-7)} = \frac{2}{7}
  \]
- Y-intercept \( b_A = 2 \)
- Equation of Line A:
  \[
  y = \frac{2}{7}x + 2
  \]

### Line B:
- Points: \( (1, 0) \) and \( (0, 1) \)
- Slope \( m_B \):
  \[
  m_B = \frac{1 - 0}{0 - 1} = -1
  \]
- Y-intercept \( b_B = 1 \)
- Equation of Line B:
  \[
  y = -x + 1
  \]

### Line C:
- Points: \( (2, 0) \) and \( (0, 1) \)
- Slope \( m_C \):
  \[
  m_C = \frac{1 - 0}{0 - 2} = -\frac{1}{2}
  \]
- Y-intercept \( b_C = 1 \)
- Equation of Line C:
  \[
  y = -\frac{1}{2}x + 1
  \]

### Line D:
- Points: \( (4, 0) \) and \( (0, 2) \)
- Slope \( m_D \):
  \[
  m_D = \frac{2 - 0}{0 - 4} = -\frac{1}{2}
  \]
- Y-intercept \( b_D = 2 \)
- Equation of Line D:
  \[
  y = -\frac{1}{2}x + 2
  \]

### Line E:
- Points: \( (0, -5) \) and \( (5, 0) \)
- Slope \( m_E \):
  \[
  m_E = \frac{0 - (-5)}{5 - 0} = \frac{5}{5} = 1
  \]
- Y-intercept \( b_E = -5 \)
- Equation of Line E:
  \[
  y = x - 5
  \]

### Final Equations:
- Line A: \( y = \frac{2}{7}x + 2 \)
- Line B: \( y = -x + 1 \)
- Line C: \( y = -\frac{1}{2}x + 1 \)
- Line D: \( y = -\frac{1}{2}x + 2 \)
- Line E: \( y = x - 5 \)

Do you want further details on any specific part of this process?

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Here are some related questions:

1. How can we verify the correctness of the slope calculations?
2. What happens to the equation if the slope is 0 or undefined?
3. How can we use these equations to find intersections between the lines?
4. How does the y-intercept affect the position of the line?
5. Can you derive the equation of a vertical line from this graph?

**Tip**: Remember, the y-intercept is always the value of \( y \) when \( x = 0 \).

Below is a graph that contains the lines for all the above equations. Find the line that represents each equation, based on the ordered pairs, and write the equation next to each letter for the line that it represents.

Find out how to determine the equations of lines based on a given graph. This problem covers key concepts in algebra, including calculating the slope, finding y-intercepts, and forming linear equations. Suitable for students in Grades 8-10.

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