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I see that you've uploaded an image of a math worksheet, which seems to focus on straight-line equations and gradients (slope). I will extract and help with solving the problems.

### Problem Overview:

1. **Question 4(a):** Find the value of \( p \) for the following:
   - (i) The gradient of a line passing through \( (-5, 7) \) and \( (p, 8) \) is 1.
   - (ii) The gradient of a line passing through \( (-5, -3) \) and \( (p, 5) \) is \( \frac{1}{2} \).
   - (iii) The intercept on the x-axis is 3, and the gradient is \( p \).

2. **Question 4(b):** In the diagram, triangle \( PQR \) is a right-angled triangle. \( PR = 10 \) units, and \( T \) is the midpoint of \( PR \).
   - You are asked to:
     - Find \( q \) (y-coordinate of point \( P(5, q) \)).
     - Find the length of \( PR \).
     - Determine the gradient of line \( TQ \).

3. **Question 5(a):** Determine the gradient of each line based on the given points.

4. **Question 5(b):** Find the gradient of the line \( PQ \) for each pair of coordinates.

5. **Question 5(c):** Solve for the unknowns in the line equations based on the provided gradients and points.

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I'll now work through some of these problems step by step. Let's start with **Question 4(a)**.

#### Solving Question 4(a):

For each part, we will use the formula for the gradient of a line between two points \( (x_1, y_1) \) and \( (x_2, y_2) \):

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

(a) 
1. **(i) Gradient = 1 for the points \( (-5, 7) \) and \( (p, 8) \):**

\[
1 = \frac{8 - 7}{p - (-5)} = \frac{1}{p + 5}
\]

Solve for \( p \):

\[
p + 5 = 1 \implies p = -4
\]

2. **(ii) Gradient \( \frac{1}{2} \) for the points \( (-5, -3) \) and \( (p, 5) \):**

\[
\frac{1}{2} = \frac{5 - (-3)}{p - (-5)} = \frac{8}{p + 5}
\]

Solve for \( p \):

\[
\frac{1}{2} = \frac{8}{p + 5} \implies p + 5 = 16 \implies p = 11
\]

3. **(iii) Gradient \( p \) and intercept on the x-axis is 3:**

The equation of a line can be written as:

\[
y = mx + c
\]

Given that the x-intercept is 3, it means the line passes through \( (3, 0) \). The equation becomes:

\[
0 = p(3) + c
\]

Also, given the y-intercept, \( c \), is the intercept on the y-axis. This part needs more clarification regarding the y-intercept value.

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Should I proceed with solving the next problems or go into more details on any of these parts?

Solve various questions on straight-line equations, including finding gradients, x-intercepts, and solving for unknown variables. Specific tasks include determining the value of p, solving for line gradients, and understanding the properties of triangle PQR in a Cartesian plane.

Learn how to solve straight-line equations and gradients through a step-by-step solution. This worksheet covers key concepts like finding the value of p, understanding gradients, midpoints, and solving geometry problems involving a right-angled triangle. Suitable for Grades 9-11.

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