Equations of Linear Graphs
3.5 Equations of Linear Graphs – Cheat Sheet
1. Forms of a Straight-Line Equation
| Form | Example | Notes |
|---|---|---|
| Gradient–Intercept form | y = mx + c | m = gradient, c = y-intercept |
| General form | ax + by = c | Rearrange to y = mx + c for gradient and intercept |
| Vertical line | x = k | Gradient undefined |
| Horizontal line | y = k | Gradient m = 0 |
2. Finding Gradient and y-Intercept from Equation
Example: 5x + 4y = 8
- Rearrange: 4y = −5x + 8 → y = (−5/4)x + 2
- Gradient m = −5/4, y-intercept = 2
3. Finding Equation from a Graph
Case 1: Given Gradient and y-Intercept
- Equation: y = mx + c
- Example: m = 2, c = −3 → y = 2x − 3
Case 2: Given Gradient and a Point
- Use y − y₁ = m(x − x₁)
- Example: m = 3, point (2, 5) → y − 5 = 3(x − 2) → y = 3x − 1
Case 3: Given Two Points
- Find m = (y₂ − y₁) / (x₂ − x₁)
- Use point–slope form y − y₁ = m(x − x₁)
- Rearrange to y = mx + c
4. Special Cases
| Line Type | Equation | Gradient |
|---|---|---|
| Horizontal | y = k | 0 |
| Vertical | x = k | Undefined |
| Through origin | y = mx | m |
5. Examples Table for Quick Reference
| Given | Working | Equation |
|---|---|---|
| m = 2, c = −3 | y = 2x − 3 | y = 2x − 3 |
| Points (1, 2) & (3, 6) | m = (6 − 2)/(3 − 1) = 2, y − 2 = 2(x − 1) | y = 2x |
| 5x + 4y = 8 | y = (−5/4)x + 2 | y = (−5/4)x + 2 |