Graph each linear equation by finding the gradient and y-intercept first.

a y 1⁄4 2x þ 1 b y 1⁄4 3x  2 c y 1⁄4 2x d y 1⁄4 x

2  1

e y 1⁄4 2x þ 3 f y 1⁄4  3x

4 g y 1⁄4 5x þ 2

2 h y 1⁄4 3x  20

To graph each linear equation, we need to express each in **slope-intercept form**:  
\[
y = mx + b
\]
Where:
- \( m \) is the **gradient (slope)**.
- \( b \) is the **y-intercept** (the point where the line crosses the y-axis).

Let's go through each equation one by one:

### (a) \( y = \frac{1}{4}(2x + 1) \)

This equation can be simplified first:

\[
y = \frac{1}{4} \times 2x + \frac{1}{4} \times 1
\]
\[
y = \frac{2}{4}x + \frac{1}{4}
\]
\[
y = \frac{1}{2}x + \frac{1}{4}
\]
- **Slope (m)** = \( \frac{1}{2} \)
- **y-intercept (b)** = \( \frac{1}{4} \)

### (b) \( y = \frac{1}{4}(3x - 2) \)

Simplifying this equation:

\[
y = \frac{1}{4} \times 3x + \frac{1}{4} \times (-2)
\]
\[
y = \frac{3}{4}x - \frac{2}{4}
\]
\[
y = \frac{3}{4}x - \frac{1}{2}
\]
- **Slope (m)** = \( \frac{3}{4} \)
- **y-intercept (b)** = \( -\frac{1}{2} \)

### (c) \( y = \frac{1}{4}(2x) \)

This equation simplifies as:

\[
y = \frac{2}{4}x
\]
\[
y = \frac{1}{2}x
\]
- **Slope (m)** = \( \frac{1}{2} \)
- **y-intercept (b)** = \( 0 \)

### (d) \( y = \frac{1}{4}x \)

Already in slope-intercept form:
- **Slope (m)** = \( \frac{1}{4} \)
- **y-intercept (b)** = \( 0 \)

### (e) \( y = \frac{1}{4}(2x + 3) \)

Simplifying this equation:

\[
y = \frac{1}{4} \times 2x + \frac{1}{4} \times 3
\]
\[
y = \frac{2}{4}x + \frac{3}{4}
\]
\[
y = \frac{1}{2}x + \frac{3}{4}
\]
- **Slope (m)** = \( \frac{1}{2} \)
- **y-intercept (b)** = \( \frac{3}{4} \)

### (f) \( y = \frac{1}{4}(-3x) \)

This equation simplifies as:

\[
y = -\frac{3}{4}x
\]
- **Slope (m)** = \( -\frac{3}{4} \)
- **y-intercept (b)** = \( 0 \)

### (g) \( y = \frac{1}{4}(5x + 2) \)

Simplifying this equation:

\[
y = \frac{1}{4} \times 5x + \frac{1}{4} \times 2
\]
\[
y = \frac{5}{4}x + \frac{2}{4}
\]
\[
y = \frac{5}{4}x + \frac{1}{2}
\]
- **Slope (m)** = \( \frac{5}{4} \)
- **y-intercept (b)** = \( \frac{1}{2} \)

### (h) \( y = \frac{1}{4}(3x - 20) \)

Simplifying this equation:

\[
y = \frac{1}{4} \times 3x + \frac{1}{4} \times (-20)
\]
\[
y = \frac{3}{4}x - \frac{20}{4}
\]
\[
y = \frac{3}{4}x - 5
\]
- **Slope (m)** = \( \frac{3}{4} \)
- **y-intercept (b)** = \( -5 \)

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Would you like to see these graphs? Or if you'd like more details, feel free to ask! Here's a set of relative questions:

1. What is the general formula for the slope-intercept form?
2. How do you calculate the slope given two points on a line?
3. How does the y-intercept help in graphing the equation?
4. Can the slope be negative? What does a negative slope mean?
5. What does it mean when the y-intercept is zero?

**Tip**: When graphing lines, always start by plotting the y-intercept and then use the slope to find other points on the line.

Graph each linear equation by finding the gradient and y-intercept first. a y 1⁄4 2x + 1, b y 1⁄4 3x - 2, c y 1⁄4 2x, d y 1⁄4 x, e y 1⁄4 2x + 3, f y 1⁄4 -3x, g y 1⁄4 5x + 2, h y 1⁄4 3x - 20

Learn how to graph linear equations by finding the gradient and y-intercept. This page covers a variety of equations in slope-intercept form, helping students understand key algebraic concepts, including how to identify slopes and y-intercepts. Suitable for students in Grades 8-10.

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