Algebra and Linear Equations: Year 7 Study Guide
Algebra uses letters to stand for numbers, so one rule can describe many situations.
A variable is a letter that represents a number. An expression such as 4n + 3 describes a rule: for n = 5 it gives 23. An equation says two expressions are equal and can be solved to find the unknown. Terms with the same variable part are like terms and can be combined: 5a + 2b − 3a = 2a + 2b.
A linear relationship has a constant rate of change and graphs as a straight line. The rule y = 2x + 1 gives the points (0, 1), (1, 3), (2, 5) and so on.
Simplifying and expanding
To expand, multiply each term inside brackets: 3(a + 4) = 3a + 12 and −2(x − 5) = −2x + 10. To factorise, take out the common factor: 6x + 9 = 3(2x + 3). Collect like terms after expanding: 2(x + 3) + 5x = 7x + 6.
| Expression | Simplified | Rule used |
|---|---|---|
| 3a + 5a | 8a | like terms |
| 4(x − 2) | 4x − 8 | expand |
| 10m − 6 | 2(5m − 3) | factorise |
| 2(p + 1) + 3p | 5p + 2 | expand and collect |
Solving equations
Keep the equation balanced: do the same to both sides. For 3x + 7 = 25, subtract 7 to get 3x = 18, then divide by 3 to get x = 6. For equations with brackets, expand first: 2(y − 4) = 10 → 2y − 8 = 10 → y = 9. Check by substitution. Unknowns on both sides: 5x − 3 = 2x + 9 → 3x = 12 → x = 4.
If needed.
And numbers on the other.
To isolate the variable.
Substitute in the original.
Patterns and graphs
A table of values from y = 3x − 1 gives (1, 2), (2, 5), (3, 8). The graph is a straight line; the gradient is the rise over run (3), and it crosses the y-axis at −1. Patterns in matchstick shapes can be described by a rule; check by substituting a term number.
Worked examples and common mistakes
Example: A tradie charges $60 plus $45 per hour: C = 45h + 60. If the bill is $285: 45h = 225, h = 5. Common mistakes: only expanding the first term, adding unlike terms (3x + 2y ≠ 5xy) and doing an operation on one side only. Write each step on a new line.
Check your understanding: model answers
1. 5x + 6. 2. m = 6. 3. y = 9.
✨ Key Takeaways
- Combine like terms only.
- Expand every term in the brackets.
- Do the same to both sides.
- Linear relationships graph as straight lines.
💬 Questions for Discussion & Reflection
- Expand and simplify 3(x + 2) + 2x.
- Solve 4m − 5 = 19.
- Find y when x = 4 in y = 2x + 1.
Print coordinate plane grids →