5.2 Equations of Straight Lines
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Learning Objectives
- Understand the forms of straight-line equations, including y = mx + c and y - y₁ = m(x - x₁)
- Find the equation of a line given a point and a gradient
- Find the equation of a line through two points
- Understand and use perpendicular and parallel gradients
- Apply straight-line equations in coordinate geometry problems
Key Concepts
Forms of Straight-Line Equations
- The most common form is **y = mx + c**, where:
- **m** is the gradient
- **c** is the y-intercept
- Another useful form is the **point-gradient form**: **y - y₁ = m(x - x₁)**
- This is used when you're given a point and a gradient.
Finding the Equation Given a Point and Gradient
- Use **y - y₁ = m(x - x₁)** if you know a point (x₁, y₁) and the gradient m.
- Rearranging gives the equation in the y = mx + c form.
- Example: A line with gradient 2 through point (3, 4):
→ y - 4 = 2(x - 3) → y = 2x - 2
Finding the Equation Through Two Points
- Step 1: Find the gradient m using **(y₂ - y₁)/(x₂ - x₁)**
- Step 2: Use one of the points in **y - y₁ = m(x - x₁)**
- Example: Find the equation through (1, 2) and (3, 6):
- Gradient = (6 - 2)/(3 - 1) = 2
- Use point (1, 2): y - 2 = 2(x - 1) → y = 2x
Parallel and Perpendicular Lines
- **Parallel lines** have the **same gradient**.
- **Perpendicular lines** have gradients that multiply to **-1**.
- If a line has gradient m, a perpendicular line has gradient **-1/m**.
- Example: A line perpendicular to y = 3x + 1 has gradient **-1/3**.
Real-World and Geometric Contexts
- Equations of lines are used to:
- Solve intersection problems
- Work with shapes and angles in coordinate geometry
- Model linear relationships in physics or economics
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