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This graph shows several lines labeled (a) through (g), and the task is to determine the equations of these lines. The general form for a linear equation is:

\[
y = mx + b
\]

Where:
- \( m \) is the slope of the line (how steep it is).
- \( b \) is the y-intercept (the value of \( y \) when \( x = 0 \)).

To find the equations for each line, we will:
1. Calculate the slope \( m \) of each line, which is \( m = \frac{\Delta y}{\Delta x} \) (rise over run).
2. Identify the y-intercept \( b \), which is where the line crosses the y-axis.

Let's analyze each line one by one:

### Line (a) - Red Line:
- This line crosses the y-axis at \( y = 6 \) (so, \( b = 6 \)).
- To find the slope, pick two points. From the graph:
  - Point 1: \( (0, 6) \)
  - Point 2: \( (6, 0) \)
  Slope \( m = \frac{6 - 0}{0 - 6} = -1 \).
- The equation is: \( y = -x + 6 \).

### Line (b) - Pink Line:
- The line crosses the y-axis at \( y = -2 \) (so, \( b = -2 \)).
- For slope, pick two points:
  - Point 1: \( (0, -2) \)
  - Point 2: \( (4, 2) \)
  Slope \( m = \frac{-2 - 2}{0 - 4} = 1 \).
- The equation is: \( y = x - 2 \).

### Line (c) - Black Line:
- The y-intercept is \( y = 0 \) (passes through the origin).
- For slope, pick two points:
  - Point 1: \( (0, 0) \)
  - Point 2: \( (4, 8) \)
  Slope \( m = \frac{8 - 0}{4 - 0} = 2 \).
- The equation is: \( y = 2x \).

### Line (d) - Blue Line:
- The y-intercept is \( y = -2 \) (so, \( b = -2 \)).
- For slope, pick two points:
  - Point 1: \( (0, -2) \)
  - Point 2: \( (-2, -10) \)
  Slope \( m = \frac{-2 + 10}{0 + 2} = 4 \).
- The equation is: \( y = 4x - 2 \).

### Line (e) - Orange Line:
- The y-intercept is \( y = -2 \).
- For slope, pick two points:
  - Point 1: \( (0, -2) \)
  - Point 2: \( (-2, 0) \)
  Slope \( m = \frac{-2 - 0}{0 + 2} = -1 \).
- The equation is: \( y = -x - 2 \).

### Line (f) - Brown Line:
- The y-intercept is \( y = 3 \).
- For slope, pick two points:
  - Point 1: \( (0, 3) \)
  - Point 2: \( (3, 0) \)
  Slope \( m = \frac{3 - 0}{0 - 3} = -1 \).
- The equation is: \( y = -x + 3 \).

### Line (g) - Green Line:
- The y-intercept is \( y = 4 \).
- For slope, pick two points:
  - Point 1: \( (0, 4) \)
  - Point 2: \( (2, 0) \)
  Slope \( m = \frac{4 - 0}{0 - 2} = -2 \).
- The equation is: \( y = -2x + 4 \).

### Summary of Equations:
- (a): \( y = -x + 6 \)
- (b): \( y = x - 2 \)
- (c): \( y = 2x \)
- (d): \( y = 4x - 2 \)
- (e): \( y = -x - 2 \)
- (f): \( y = -x + 3 \)
- (g): \( y = -2x + 4 \)

Would you like more details on any specific line or have any other questions?

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Here are some related questions to expand your understanding:
1. How do you determine the slope of a line given two points?
2. What happens when the slope of a line is zero?
3. How does the y-intercept affect the graph of a line?
4. Can two lines have the same slope but different y-intercepts? What does that mean geometrically?
5. How would you calculate the x-intercept of a line?

**Tip**: To quickly find the slope of a line, always remember the formula \( m = \frac{\Delta y}{\Delta x} \) (rise over run).

Bestem funksjonsuttrykkene til linjene (a)-(g) i figuren over.

Solve for the linear equations of lines labeled (a)-(g) in the provided graph using slope-intercept form. This problem covers key algebraic concepts like calculating slope and identifying y-intercepts. Ideal for students in Grades 8-10.

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