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### **Question 1: Write down the gradient of each of these lines.**

Recall that the gradient (or slope) of a straight line in the form \( y = mx + c \) is the coefficient \( m \) of \( x \).

- **(a) \( y = 3x + 1 \):** The gradient is \( 3 \).
- **(b) \( y = 2x - 5 \):** The gradient is \( 2 \).
- **(c) \( y = 7x + 4 \):** The gradient is \( 7 \).
- **(d) \( y = 10x + 5 \):** The gradient is \( 10 \).
- **(e) \( y = x - 2 \):** The gradient is \( 1 \) (since the coefficient of \( x \) is implicitly 1).
- **(f) \( y = 6x \):** The gradient is \( 6 \).
- **(g) \( y = -4x + 3 \):** The gradient is \( -4 \).
- **(h) \( y = -3x - 7 \):** The gradient is \( -3 \).
- **(i) \( y = \frac{1}{2}x + 3 \):** The gradient is \( \frac{1}{2} \).
- **(j) \( y = -\frac{4}{5}x - 9 \):** The gradient is \( -\frac{4}{5} \).

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### **Question 2: Write down where each of these lines cross the y-axis (y-intercept).**

The y-intercept is the value of \( y \) when \( x = 0 \). In the equation \( y = mx + c \), the y-intercept is \( c \).

- **(a) \( y = 2x + 3 \):** The y-intercept is \( 3 \).
- **(b) \( y = 7x + 1 \):** The y-intercept is \( 1 \).
- **(c) \( y = 3x - 2 \):** The y-intercept is \( -2 \).
- **(d) \( y = x - 5 \):** The y-intercept is \( -5 \).
- **(e) \( y = 2x \):** The y-intercept is \( 0 \) (since there's no constant term).
- **(f) \( y = -4x + 6 \):** The y-intercept is \( 6 \).
- **(g) \( y = -5x - 3 \):** The y-intercept is \( -3 \).
- **(h) \( y = -3x \):** The y-intercept is \( 0 \).
- **(i) \( y = \frac{4}{3}x + \frac{2}{5} \):** The y-intercept is \( \frac{2}{5} \).
- **(j) \( y = -\frac{2}{3}x - \frac{1}{2} \):** The y-intercept is \( -\frac{1}{2} \).

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### **Question 3: Write down the equation of the lines below.**

The general form of a line is \( y = mx + c \), where \( m \) is the gradient and \( c \) is the y-intercept.

- **(a)** Gradient = 3, y-intercept = 6: The equation is \( y = 3x + 6 \).
- **(b)** Gradient = 2, y-intercept = -1: The equation is \( y = 2x - 1 \).
- **(c)** Gradient = -4, y-intercept = 3: The equation is \( y = -4x + 3 \).
- **(d)** Gradient = 8, y-intercept = 4: The equation is \( y = 8x + 4 \).
- **(e)** Gradient = 1, passing through (0, 4): The equation is \( y = x + 4 \) (since the y-intercept is 4).
- **(f)** Passing through (0, -2), gradient 4: The equation is \( y = 4x - 2 \).
- **(g)** Gradient = -5, passing through the origin (0, 0): The equation is \( y = -5x \) (since the y-intercept is 0).

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### **Question 4: Does the point (2, 5) lie on the line \( y = 3x - 1 \)?**

To check if the point lies on the line, substitute \( x = 2 \) into the equation and see if it results in \( y = 5 \):

For the line \( y = 3x - 1 \):

\[
y = 3(2) - 1 = 6 - 1 = 5
\]

Yes, the point \( (2, 5) \) lies on the line \( y = 3x - 1 \).

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Would you like further explanations on any of the questions, or should we dive deeper into specific concepts? Here are 5 related questions to expand on this topic:

1. How do you derive the equation of a line if given two points?
2. What is the significance of the gradient in real-life scenarios?
3. How can you determine if two lines are parallel or perpendicular?
4. What happens to the graph of a line when the gradient increases or decreases?
5. How can you find the x-intercept of a line from its equation?

**Tip:** The gradient represents how steep a line is; a higher gradient means a steeper incline.

Solve mathematical problems related to gradients, y-intercepts, line equations, and point checks.

This page covers step-by-step solutions for finding gradients, y-intercepts, and forming line equations. It also checks whether a given point lies on a line, ideal for students in Grades 8-10.

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