https://file.aiquickdraw.com/mathbot/file/1729757245873tnie3sv4.jpg

Let's work through the questions based on the provided diagram:

### (a) Write down the coordinates of point \( P \).

From the diagram, point \( P \) appears to be located at \((-3, -2)\). Hence:
- Coordinates of point \( P \) are \((-3, -2)\).

### (b) Find the gradient of line \( L \).

The gradient (slope) \( m \) of a line is calculated as the change in \( y \)-coordinates divided by the change in \( x \)-coordinates, i.e.,

\[
m = \frac{{y_2 - y_1}}{{x_2 - x_1}}
\]

From the graph, two clear points on line \( L \) are:
- Point 1: \( (0, -1) \) (the y-intercept)
- Point 2: \( (3, 2) \)

Using these points:
\[
m = \frac{{2 - (-1)}}{{3 - 0}} = \frac{3}{3} = 1
\]

Thus, the gradient of line \( L \) is \( 1 \).

### (c) Write down the equation of line \( L \) in the form \( y = mx + c \).

The equation of a line in slope-intercept form is \( y = mx + c \), where:
- \( m \) is the gradient (which we've found to be \( 1 \)).
- \( c \) is the y-intercept.

From the graph, the y-intercept is \( -1 \). Hence, the equation of the line is:
\[
y = 1x - 1 \quad \text{or simply} \quad y = x - 1
\]

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Do you need further clarification on any step or have questions? Here are some additional questions related to this problem:

1. What does the gradient represent geometrically?
2. Can a line have a negative gradient, and what does it mean?
3. How would the equation change if the gradient were different?
4. What are the conditions for two lines to be parallel?
5. How can you find the equation of a line passing through two given points?

**Tip**: When finding the gradient from a graph, always pick two clear points with integer coordinates to minimize errors.

The diagram shows a point P and a line L. (a) Write down the coordinates of point P. (b) Find the gradient of line L. (c) Write down the equation of line L in the form y = mx + c.

Solve a coordinate geometry problem by finding the coordinates of point P, calculating the gradient of line L, and determining its equation in the form y = mx + c. Learn key concepts like slope and linear equations. Suitable for Grades 7-9.

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