Coordinate Geometry: Equations of Linear Graphs (0607)

Welcome to the world of Linear Graphs! This chapter is your key to unlocking how straight lines behave mathematically. Think of it like learning the address and direction of every straight road on a giant coordinate map. Understanding these equations is essential not just for graphing, but also for solving real-world problems involving constant rates of change, like speed or simple interest.

Section 1: The Core Equation - Slope-Intercept Form

1.1 The Master Formula: \(y = mx + c\)

Every non-vertical straight line can be perfectly described by this famous equation. Make sure you know what each letter stands for:

  • \(x\) and \(y\): These are the coordinates of any point lying on the line. They are the variables that change as you move along the graph.
  • \(m\): The Gradient (or slope). This tells you how steep the line is and its direction.
  • \(c\): The Y-Intercept. This is the coordinate where the line crosses the vertical y-axis (the point \((0, c)\)).

Quick Review Box: The roles of \(m\) and \(c\)

\(m\) (Gradient): Tells you how much \(y\) changes for every 1 unit change in \(x\).
\(c\) (Y-Intercept): Tells you where the line hits the y-axis.

1.2 Understanding the Gradient (\(m\))

The gradient (\(m\)) is a measure of steepness. We often remember it as "Rise over Run."

Imagine walking up a hill (the line). The gradient is the ratio of how much you go up (the rise, or change in \(y\)) compared to how much you go across (the run, or change in \(x\)).

Calculating Gradient from Two Points (E4.2)

If you have two points on the line, \((x_1, y_1)\) and \((x_2, y_2)\), the gradient \(m\) is calculated using the formula:

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

Types of Gradient:

  • Positive Gradient (\(m > 0\)): The line slopes upwards from left to right (like a normal hill).
  • Negative Gradient (\(m < 0\)): The line slopes downwards from left to right.
  • Zero Gradient (\(m = 0\)): The line is perfectly horizontal.
  • Undefined Gradient: The line is perfectly vertical (you cannot divide by zero).

Common Mistake to Avoid!
Always keep your coordinates in the same order! If you start with \(y_2\) in the numerator, you must start with \(x_2\) in the denominator.

Section 2: Finding the Equation of a Straight Line

The main goal is always to find the values of \(m\) and \(c\) and substitute them back into \(y = mx + c\).

2.1 Case 1: Given the Gradient (\(m\)) and the Y-Intercept (\(c\))

This is the easiest case! You just plug the numbers straight into \(y = mx + c\).

Example: If the gradient is 5 and the y-intercept is -2, the equation is \(y = 5x - 2\).

2.2 Case 2: Given Two Points \((x_1, y_1)\) and \((x_2, y_2)\)

Step 1: Calculate \(m\).
Use the gradient formula: \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

Step 2: Substitute \(m\) and one point into \(y = mx + c\) to find \(c\).
Pick either \((x_1, y_1)\) or \((x_2, y_2)\) (it doesn't matter which one!) and plug the values for \(x\), \(y\), and your calculated \(m\) into the equation. Rearrange to find \(c\).

Step 3: Write the final equation.
Substitute \(m\) and \(c\) back into \(y = mx + c\).

Step-by-Step Example
Find the equation of the line passing through \((2, 10)\) and \((5, 1)\).

1. Find \(m\):
\(m = \frac{1 - 10}{5 - 2} = \frac{-9}{3} = -3\)
So far: \(y = -3x + c\)

2. Find \(c\): (Use point \((2, 10)\), so \(x=2\) and \(y=10\))
\(10 = -3(2) + c\)
\(10 = -6 + c\)
\(c = 16\)

3. Final Equation: \(y = -3x + 16\)

2.3 Other Forms of Linear Equations (E4.4)

While \(y = mx + c\) is the most useful form for graphing and finding gradients, sometimes lines are written in the General Form: \(ax + by = c\).

If you see a line in this format, you must rearrange it into \(y = mx + c\) to easily identify the gradient and y-intercept.

Example: Convert \(5x + 4y = 8\) into slope-intercept form (E4.4).

1. Isolate the \(y\) term:
\(4y = 8 - 5x\)

2. Divide by the coefficient of \(y\):
\(y = \frac{8}{4} - \frac{5}{4}x\)
\(y = 2 - \frac{5}{4}x\)

3. Rearrange to \(y = mx + c\) format:
\(y = -\frac{5}{4}x + 2\)
Gradient \(m = -1.25\), Y-intercept \(c = 2\).

Section 3: Special Cases: Horizontal and Vertical Lines (C4.4/E4.4)

Not all lines fit neatly into the \(y = mx + c\) structure, particularly lines that are perfectly flat or perfectly upright.

3.1 Horizontal Lines (\(y = k\))

A horizontal line has a gradient of \(m = 0\).

Since \(y = 0x + c\), the equation simplifies to \(y = c\) (or using the syllabus notation, \(y = k\)).

Example: The line passing through \((3, 5)\) and \((9, 5)\) is \(y = 5\). The \(y\) value is always 5.

3.2 Vertical Lines (\(x = k\))

A vertical line has an Undefined Gradient.

The equation is simply \(x = k\), where \(k\) is the constant x-coordinate the line passes through.

Example: The line passing through \(( -4, 1)\) and \(( -4, 7)\) is \(x = -4\). The \(x\) value is always -4.

Section 4: Relationships Between Lines

If you compare the gradients of two different lines, you can immediately tell how they relate to each other.

4.1 Parallel Lines (C4.5 / E4.5)

Two lines are parallel if they travel in exactly the same direction and never intersect. Mathematically, this means they have the same gradient.

If Line 1 has gradient \(m_1\) and Line 2 has gradient \(m_2\):

$$ \text{Parallel Condition: } m_1 = m_2 $$

Step-by-Step Example (E4.5)
Find the equation of the line parallel to \(y = 4x - 1\) that passes through \((1, -3)\).

1. Identify the gradient: The existing line has \(m = 4\). Since the new line is parallel, its gradient is also \(m = 4\).
So far: \(y = 4x + c\)

2. Find \(c\): Substitute the point \((1, -3)\).
\(-3 = 4(1) + c\)
\(-3 = 4 + c\)
\(c = -7\)

3. Final Equation: \(y = 4x - 7\)

4.2 Perpendicular Lines (E4.6 - Extended Content Only)

Two lines are perpendicular if they intersect at a 90° angle (a right angle).

If Line 1 has gradient \(m_1\), the gradient of a line perpendicular to it, \(m_2\), must be the negative reciprocal.

$$ \text{Perpendicular Condition: } m_1 \times m_2 = -1 \quad \text{ or } \quad m_2 = -\frac{1}{m_1} $$

Memory Trick: The "Flip and Switch" Rule
To find the perpendicular gradient, you:
1. Flip the fraction (take the reciprocal).
2. Switch the sign (make it negative, or positive if it was already negative).

Example Gradients:
If \(m_1 = 3\), then \(m_2 = -\frac{1}{3}\)
If \(m_1 = -\frac{2}{5}\), then \(m_2 = +\frac{5}{2}\)

Step-by-Step Example (E4.6)
Find the equation of the line perpendicular to \(2y = 3x + 1\) that passes through the origin \((0, 0)\).

1. Find \(m_1\) (the gradient of the given line): First, rearrange into \(y = mx + c\):
\(y = \frac{3}{2}x + \frac{1}{2}\)
So, \(m_1 = \frac{3}{2}\).

2. Find \(m_2\) (the perpendicular gradient): Apply the negative reciprocal rule.
\(m_2 = -\frac{2}{3}\)
So far: \(y = -\frac{2}{3}x + c\)

3. Find \(c\): Substitute the point \((0, 0)\) (the origin).
\(0 = -\frac{2}{3}(0) + c\)
\(c = 0\)

4. Final Equation: \(y = -\frac{2}{3}x\)

Key Takeaways for Equations of Linear Graphs

Coordinate geometry combines algebra and geometry. The straight line is the simplest relationship, and mastering it means mastering the roles of \(m\) and \(c\).

Coordinate Geometry Tools (C4.3/E4.3 Reminder)

While equations are the focus, remember that you also need these tools for line segments (E4.3):

1. Length (Distance) Formula: Used to find the distance between two points \((x_1, y_1)\) and \((x_2, y_2)\). (This is based on Pythagoras' theorem!)

$$ \text{Length} = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} $$

2. Midpoint Formula: Used to find the exact middle point of the line segment.

$$ \text{Midpoint} = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right) $$

Checklist Summary
  • Identify \(m\) and \(c\) from \(y = mx + c\).
  • Calculate gradient \(m\) using \(m = \frac{y_2 - y_1}{x_2 - x_1}\).
  • Know the equation forms for horizontal lines (\(y=k\)) and vertical lines (\(x=k\)).
  • Parallel lines have the same gradient (\(m_1 = m_2\)).
  • (Extended) Perpendicular lines have negative reciprocal gradients (\(m_1 = -1/m_2\)).
  • Always give your final answer in a fully simplified form (usually \(y = mx + c\) or \(ax + by = c\), depending on the question's requirement).