Algebra
Algebraic Notation and Simplifying Expressions
Year 7 · Year 8 · Year 9
- ✓Understand and use basic algebraic notation, including variables, coefficients, and terms.
- ✓Identify and collect like terms to simplify algebraic expressions.
- ✓Multiply algebraic terms, including those with coefficients and powers.
- ✓Divide algebraic terms, including those with coefficients and powers.
- ✓Simplify expressions involving a combination of multiplication, division, and collecting like terms.
Key concepts
Algebra uses letters, called variables, to represent unknown numbers. An algebraic expression is a combination of numbers, variables, and operation signs (+, -, ×, ÷). A 'term' is a single number, a single variable, or a product of numbers and variables. A 'coefficient' is the number multiplied by a variable (e.g., in 5x, 5 is the coefficient).
Like terms are terms that have exactly the same variables raised to the same powers. The numerical coefficients can be different. For example, 3x and 5x are like terms, but 3x and 5x² are not. Similarly, 2ab and 7ab are like terms, but 2ab and 7a are not.
To simplify an expression by collecting like terms, you add or subtract their coefficients. The variable part of the term remains unchanged. Remember to include the sign in front of each term when identifying it.
To multiply algebraic terms, multiply the coefficients together and then multiply the variables together. When multiplying variables with the same base, add their powers (e.g., x² × x³ = x⁵).
To divide algebraic terms, divide the coefficients together and then divide the variables together. When dividing variables with the same base, subtract their powers (e.g., x⁵ ÷ x² = x³).
Key facts to remember
- 1A variable is a letter representing an unknown number.
- 2A coefficient is the number multiplying a variable (e.g., in 5x, 5 is the coefficient).
- 3Like terms have the exact same variables raised to the exact same powers (e.g., 3x and -7x are like terms, but 3x and 3x² are not).
- 4When collecting like terms, only add or subtract their coefficients; the variable part stays the same.
- 5When multiplying terms, multiply the coefficients and add the powers of the same variables (e.g., x² × x³ = x⁵).
- 6When dividing terms, divide the coefficients and subtract the powers of the same variables (e.g., x⁵ ÷ x² = x³).
- 7A variable written without a coefficient (e.g., x) has a coefficient of 1 (i.e., 1x).
- 8A variable written without a power (e.g., x) has a power of 1 (i.e., x¹).
Worked examples
Example 1
Simplify 4a + 7b - 2a + 3b
Answer
2a + 10b
Remember to take the sign in front of the term with it.
Example 2
Simplify 5x - 3y + 2x² + y - 8x
Answer
2x² - 3x - 2y
It is good practice to write terms with higher powers first, then in alphabetical order.
Example 3
Simplify 3p × 5p²q
Answer
15p³q
Remember that p is the same as p¹.
Example 4
Simplify (12a⁵b²) / (4a²b)
Answer
3a³b
b¹ is usually written simply as b.
Example 5
Simplify 2(3x + 4y) + 5x - y
Answer
11x + 7y
Always expand brackets before collecting like terms.
Common mistakes
- ✗Adding or subtracting terms that are not 'like terms' (e.g., incorrectly simplifying 3x + 2y to 5xy).
- ✗Incorrectly handling negative signs when collecting terms (e.g., 5x - 8x = 3x instead of -3x).
- ✗Forgetting to multiply or divide the numerical coefficients when multiplying or dividing terms.
- ✗Multiplying powers instead of adding them when multiplying variables (e.g., x² × x³ = x⁶ instead of x⁵).
- ✗Dividing powers instead of subtracting them when dividing variables (e.g., x⁶ ÷ x² = x³ instead of x⁴).
Exam tips
- ★Underline or circle like terms with their signs to help avoid errors when collecting.
- ★Always deal with the numerical coefficients first, then the variables, when multiplying or dividing terms.
- ★Remember the rules for powers: add powers when multiplying, subtract powers when dividing.
- ★Show all your working step-by-step, especially when expanding brackets or dealing with multiple operations.
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