Expressions, Equations and Variables: Grade 6 Study Guide
Algebra uses letters to stand for numbers and lets us describe patterns and relationships.
A variable is a letter that stands for a number. An expression such as 3n + 5 has no equals sign; an equation such as 3n + 5 = 20 does. Terms are the parts added together, a coefficient is the number multiplying a variable and a constant is a number on its own.
Exponents show repeated multiplication: 4³ = 4 × 4 × 4 = 64. Expressions are equivalent when they have the same value for every number, such as 2(x + 3) and 2x + 6, as shown by the distributive property.
Writing expressions
Translate words to symbols. 'Five more than a number' is n + 5. 'Three times a number, decreased by 4' is 3n − 4. 'The quotient of a number and 6' is n/6. Evaluate by substituting: if n = 4, 3n − 4 = 12 − 4 = 8.
| Words | Expression |
|---|---|
| a number plus 9 | n + 9 |
| 7 less than a number | n − 7 |
| twice a number | 2n |
| the sum of x and 3, times 4 | 4(x + 3) |
Solving one-step equations
Use inverse operations to isolate the variable and keep the equation balanced. x + 8 = 15 → x = 15 − 8 = 7. 4y = 28 → y = 7. n/3 = 9 → n = 27. Check by substituting back.
What is done to the variable?
Use the inverse.
Variable alone.
Substitute into the original.
Inequalities and relationships
An inequality such as x > 3 has many solutions: any number greater than 3. It is shown on a number line with an open circle at 3 and an arrow right. In a relationship such as d = 60t (distance at 60 mph for t hours), t is the independent variable and d is the dependent variable. A table or graph shows how they change together.
Worked examples and common mistakes
Example: A gym charges a $20 fee plus $12 per class: cost = 12c + 20. With 5 classes: $80. Common mistakes: doing the operation on only one side, mixing up 3x (three times x) and x³ (x cubed) and losing a term when distributing. Substitute a number into both expressions to test equivalence.
Check your understanding: model answers
1. 13. 2. n = 12. 3. 2n − 6.
✨ Key Takeaways
- A variable stands for a number.
- Keep equations balanced.
- Use inverse operations.
- Distributive property: a(b + c) = ab + ac.
💬 Questions for Discussion & Reflection
- Evaluate 5x − 2 when x = 3.
- Solve 7 + n = 19.
- Write: six less than twice a number.