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The image shows several lines plotted on a coordinate plane with various slopes and y-intercepts. You are asked to:

1. Identify the gradient (slope) and the y-intercept for each line.
2. Write the equation of each line in the form \( y = mx + c \), where \( m \) is the gradient and \( c \) is the y-intercept.

I will walk through each line (labelled \( a \), \( b \), \( c \), \( d \), \( e \), and \( f \)) and determine their gradient and y-intercept, followed by the equations of the lines.

### Line \( a \):
- **y-intercept**: The line crosses the \( y \)-axis at \( y = 3 \).
- **Gradient**: The line slopes downward. Moving 1 unit right decreases \( y \) by 1 unit, so the slope is \( m = -1 \).
- **Equation**: \( y = -x + 3 \).

### Line \( b \):
- **y-intercept**: The line crosses the \( y \)-axis at \( y = 2 \).
- **Gradient**: Moving 1 unit right decreases \( y \) by 1 unit, so the slope is \( m = -1 \).
- **Equation**: \( y = -x + 2 \).

### Line \( c \):
- **y-intercept**: The line crosses the \( y \)-axis at \( y = 1 \).
- **Gradient**: The line is vertical, which means the gradient is undefined.
- **Equation**: The line is vertical at \( x = 0 \).

### Line \( d \):
- **y-intercept**: The line crosses the \( y \)-axis at \( y = 0 \) (the origin).
- **Gradient**: The slope is positive. Moving 1 unit right increases \( y \) by 1 unit, so the slope is \( m = 1 \).
- **Equation**: \( y = x \).

### Line \( e \):
- **y-intercept**: The line crosses the \( y \)-axis at \( y = 3 \).
- **Gradient**: The slope is positive. Moving 2 units right increases \( y \) by 3 units, so the slope is \( m = \frac{3}{2} \).
- **Equation**: \( y = \frac{3}{2}x + 3 \).

### Line \( f \):
- **y-intercept**: The line crosses the \( y \)-axis at \( y = 0 \) (the origin).
- **Gradient**: The slope is positive. Moving 1 unit right increases \( y \) by 2 units, so the slope is \( m = 2 \).
- **Equation**: \( y = 2x \).

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Let me know if you want more detailed steps or have any questions! Here are 5 relative questions to expand on this:

1. How do you find the gradient of a line from two points?
2. Can you explain why the line \( c \) has an undefined slope?
3. How do different gradients affect the steepness of lines?
4. What is the significance of the y-intercept in a line's equation?
5. How would you find the equation of a line if given only two points?

**Tip:** When calculating the gradient, always choose points that are easy to read off the graph for more accuracy.

For each graph, i) write its gradient and y-intercept, ii) write the equation of the line.

Learn how to identify the gradient and y-intercept of a line from a graph and write the equation of the line in slope-intercept form (y = mx + c). This problem covers key concepts in linear equations and is suitable for students in Grades 8-10.

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