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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 \( y = 2x + 1 \).

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**(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 \( y = \frac{1}{6}x - \frac{1}{2} \).

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**(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 \( 2x + 5y = 10 \).

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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.

Tentukan persamaan garis lurus yang: (a) Melalui titik A(1, 3) dan sejajar garis y - 2x + 1 = 0. (b) Melalui titik P(3, 0) dan tegak lurus garis 6x + y - 4 = 0. (c) Memotong sumbu x sepanjang 5 dan memotong sumbu y sepanjang 2 dari titik asal.

Learn how to determine the equations of lines through specific points, parallel to and perpendicular to given lines, and using intercepts. Includes detailed solutions with key geometry concepts such as slope, parallel and perpendicular lines, and intercept form. Suitable for Grade 10-12 students.

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