Let's work through the questions step by step:

(a) Determine the equation of a line that passes through point $A(1, 3)$ and is parallel to the line $y - 2x + 1 = 0$.

1. The given line is in the form $y - 2x + 1 = 0$, which can be rewritten as $y = 2x - 1$. The slope of this line is $m = 2$.
2. Since the required line is parallel, it will have the same slope $m = 2$.
3. Using the point-slope form of the line equation, $y - y_1 = m(x - x_1)$, where $(x_1, y_1) = (1, 3)$ and $m = 2$: $y - 3 = 2(x - 1)$ Simplifying: $y - 3 = 2x - 2$ $y = 2x + 1$ So, the equation of the line is $y = 2x + 1$.

(b) Determine the equation of a line that passes through point $P(3, 0)$ and is perpendicular to the line $6x + y - 4 = 0$.

1. Rewrite the given line $6x + y - 4 = 0$ in slope-intercept form: $y = -6x + 4$ The slope of this line is $m = -6$.
2. The slope of a line perpendicular to this will be the negative reciprocal: $m_{\text{perpendicular}} = \frac{1}{6}$.
3. Using the point-slope form again, $y - y_1 = m(x - x_1)$, with $(x_1, y_1) = (3, 0)$ and $m = \frac{1}{6}$: $y - 0 = \frac{1}{6}(x - 3)$ Simplifying: $y = \frac{1}{6}(x - 3)$ $y = \frac{1}{6}x - \frac{1}{2}$ So, the equation of the line is $y = \frac{1}{6}x - \frac{1}{2}$.

(c) Determine the equation of a line that cuts the x-axis at 5 units and the y-axis at 2 units from the origin.

1. A line that intercepts the x-axis at $(5, 0)$ and the y-axis at $(0, 2)$ can be written using the intercept form: $\frac{x}{a} + \frac{y}{b} = 1$ Here, $a = 5$ (x-intercept) and $b = 2$ (y-intercept): $\frac{x}{5} + \frac{y}{2} = 1$
2. Multiply through by 10 to clear the denominators: $2x + 5y = 10$ So, the equation of the line is $2x + 5y = 10$.

Do you want any further details or clarifications?

Here are 5 related questions to explore further:

1. How do you find the equation of a line given two points?
2. What is the slope of a vertical or horizontal line?
3. How do you determine if two lines are parallel or perpendicular just by looking at their equations?
4. Can you convert the general form of a line equation to the slope-intercept form?
5. How can you use the distance formula to verify the intercepts of a line?

Tip: The slope of a line tells you its steepness; parallel lines always have the same slope, while perpendicular lines have slopes that are negative reciprocals of each other.