International Mathematics (0607) | Parallel lines

🗺️ Coordinate Geometry: Parallel Lines – Study Notes

Hello Mathematicians! Welcome to the Coordinate Geometry chapter. This section is all about describing lines and shapes using numbers on a graph. Today, we focus on lines that run side-by-side forever: Parallel Lines.

Understanding parallel lines is crucial not just for solving exam problems, but also for real-world applications, like engineering, architecture, and even map design! Don't worry if this seems tricky at first; we will break down the rules into simple steps.

Section 1: Quick Review of Straight Lines (The Essentials)

Before we dive into parallel lines, we must remember the star of the show: the equation of a straight line.

The Equation: \(y = mx + c\)

Every straight line (except vertical lines) can be written in this form, where:

• \(m\) is the Gradient (or slope). This tells you how steep the line is and which direction it moves.
• \(c\) is the y-intercept. This is the point where the line crosses the y-axis (the coordinates are \((0, c)\)).

Understanding the Gradient (\(m\))

The gradient is a measure of steepness, often calculated as "rise over run".

\(m = \frac{\text{Change in y}}{\text{Change in x}} = \frac{y_2 - y_1}{x_2 - x_1}\)

Example: If a line goes up 3 units for every 1 unit it moves right, the gradient \(m\) is 3. If it goes down 2 units for every 5 units it moves right, the gradient \(m\) is \(-\frac{2}{5}\).

Section 2: The Golden Rule for Parallel Lines

What makes lines parallel in coordinate geometry? It's simple: They must have the exact same steepness!

Definition: What is Parallel?

Two lines are parallel if they lie in the same plane and will never intersect (cross), no matter how far they are extended.

The Parallel Line Rule

If line 1 has gradient \(m_1\) and line 2 has gradient \(m_2\), then for the lines to be parallel:

\(m_1 = m_2\)

Memory Aid: Think of two identical ski slopes running side-by-side. If one slope has a gradient of 4, the other must also have a gradient of 4, or they will eventually crash into each other or separate!

Did You Know?

In mathematics, the symbol used to show that line A is parallel to line B is two vertical lines: \(A \parallel B\).

Section 3: Finding the Equation of a Parallel Line (Step-by-Step)

This is the most common exam question for this topic. You are usually given the equation of one line and a single point that the new parallel line must pass through.

Example Problem:

Find the equation of the line that is parallel to the line \(y = 4x - 1\) and passes through the point \((1, -3)\). (This is exactly the type of example given in the syllabus!)

Follow these four steps carefully:

Step 1: Determine the gradient of the given line.

• The given line is \(y = 4x - 1\).
• Since it is already in the form \(y = mx + c\), we can easily see the gradient, \(m\).
• Gradient of the given line, \(m_1 = 4\).

Step 2: State the gradient of the new parallel line.

• Since the new line is parallel, it must have the same gradient.
• Gradient of the new line, \(m_2 = 4\).

Step 3: Use the new gradient and the given point to find the y-intercept (\(c\)).

• We know the new line's equation looks like: \(y = 4x + c\).
• We also know the line must pass through the point \((1, -3)\). This means when \(x = 1\), \(y = -3\).
• Substitute these values into the equation:
  \(y = 4x + c\)
  \(-3 = 4(1) + c\)
  \(-3 = 4 + c\)
  \(-3 - 4 = c\)
  \(c = -7\)

Step 4: Write the final equation.

• Now that we have the gradient (\(m=4\)) and the y-intercept (\(c=-7\)), we write the final equation in the standard \(y = mx + c\) form.
• The equation of the parallel line is: \(y = 4x - 7\).

What if the Equation Isn't in \(y = mx + c\) Form?

Sometimes the original equation looks like \(Ax + By = C\). You must rearrange it first!

Example: Find the gradient of a line parallel to \(5x + 2y = 10\).

1. Rearrange to isolate \(y\):
   \(2y = -5x + 10\)
2. Divide by 2:
   \(y = -\frac{5}{2}x + 5\)
3. The gradient is \(m = -\frac{5}{2}\). The parallel line will also have a gradient of \(-\frac{5}{2}\).

Section 4: Avoiding Mistakes and Final Summary

🚫 Common Mistakes to Avoid

These are the small errors that cost marks in the exam:

1. Not Rearranging: Forgetting to rearrange \(Ax + By = C\) into \(y = mx + c\). You cannot just take the coefficient of \(x\) if \(y\) is not isolated!
2. Sign Errors: When substituting the coordinates \((x, y)\) in Step 3, be very careful with negative numbers.
3. Using the Wrong Intercept: The new line must pass through the new point. Students sometimes mistakenly use the \(c\) from the original line instead of calculating a new \(c\).