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: $y = 2x$.

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.