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) = $-5$

---

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.