Linear Equations, Slope and Functions: Grade 8 Study Guide
Linear relationships describe steady change, such as a taxi fare or a phone plan.
A linear equation graphs as a straight line. The slope m measures steepness: rise ÷ run, or (y₂ − y₁) ÷ (x₂ − x₁). The y-intercept b is where the line crosses the y-axis. In slope-intercept form, y = mx + b. For y = 2x + 3, the slope is 2 and the line crosses the y-axis at (0, 3).
A function assigns exactly one output to each input. A graph is a function if no vertical line crosses it more than once. A linear function has a constant rate of change; in real contexts, the slope is the rate (cost per hour) and the intercept is the starting value (a fixed fee).
Finding slope from two points
For (1, 3) and (4, 12): slope = (12 − 3) ÷ (4 − 1) = 9 ÷ 3 = 3. A positive slope rises left to right, a negative slope falls, zero slope is horizontal and an undefined slope is vertical.
| Points | Rise | Run | Slope |
|---|---|---|---|
| (0, 1), (2, 5) | 4 | 2 | 2 |
| (1, 6), (3, 2) | −4 | 2 | −2 |
| (0, 4), (5, 4) | 0 | 5 | 0 |
Writing an equation
A gym charges a $25 sign-up fee plus $15 per month. Cost y after x months: y = 15x + 25. The slope 15 is the monthly rate; 25 is the starting fee. To write an equation from a graph, read the y-intercept and use two points for the slope.
Rise over run.
Where x = 0.
Substitute m and b.
Does it fit?
Multi-step equations and functions
Solve 3(x − 2) + 4 = 2x + 7: 3x − 6 + 4 = 2x + 7 → 3x − 2 = 2x + 7 → x = 9. An equation can have one solution, no solution (3x + 1 = 3x + 5) or infinitely many (2x + 2 = 2(x + 1)). Compare two functions given in different forms (a table and an equation) by comparing slopes and intercepts.
Worked examples and common mistakes
Example: A plant is 4 cm tall and grows 2 cm per week: h = 2w + 4. After 6 weeks, h = 16 cm. Common mistakes: swapping rise and run, mixing up x and y in the slope formula and ignoring the units. Always interpret the slope in the story.
Check your understanding: model answers
1. 8 ÷ 4 = 2. 2. y = 3x − 2. 3. 3x = 15, so x = 5.
✨ Key Takeaways
- Slope = rise ÷ run.
- y = mx + b: m is the slope and b the y-intercept.
- A function has one output for each input.
- Interpret slope as a rate.
💬 Questions for Discussion & Reflection
- Find the slope through (2, 1) and (6, 9).
- Write the equation: slope 3, y-intercept −2.
- Solve 5x − 4 = 2x + 11.
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