Algebra Toolkit

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Lesson notes

Algebraic Notation

Algebra uses letters to represent general or unknown numbers; letters are called variables.
Multiplication is written without a symbol: ab means a × b, 3ab means 3 × a × b.
Division is written as a fraction: a/b means a ÷ b.
Powers (indices) and roots work as with numbers: a² means a × a, √a means square root of a.
Brackets indicate the operation inside is done first: 3(a + b) means 3 × (a + b).
Order of operations (BIDMAS) applies to algebra just as with numbers.

Algebraic Vocabulary

A term is a variable, a number (constant), or a product of numbers and variables (e.g., 5x, 4xy²).
The coefficient is the number in front of a variable: in -5y, the coefficient is -5.
A factor divides a term exactly: factors of 3x are 1, 3, x, and 3x.
An expression has no equals sign (e.g., 2x + 5y).
An equation has an equals sign and can be solved (e.g., 2x = 10).
A formula is a rule relating quantities (e.g., P = 2l + 2w); substituting values turns it into an equation.

Substitution

Substitution means replacing a variable with a given number.
Always use brackets around negative numbers when substituting: if x = -4, then x² = (-4)² = 16.
Follow the order of operations (BIDMAS) carefully after substitution.
Substitution can produce an equation to solve (e.g., 20 = 2l + 8 gives l = 6).
On a calculator, put brackets around any substituted negative numbers.

Collecting Like Terms

Like terms have exactly the same variables and powers (e.g., 2x and 3x; 5xy and -7xy).
Unlike terms have different variables or powers (e.g., 2x and 3y; 4x² and 6x).
Collect like terms by adding or subtracting their coefficients: 2x + 3x = 5x; 4y - 10y = -6y.
The sign in front of a term belongs to that term; terms can be reordered with their signs.
Simplify coefficients of 1 or -1 to just the variable (e.g., 1x → x, -1y → -y).
Collect different types of like terms separately: 2x + 4y + 5x - 3y = 7x + y.

Slides

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Practice questions

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1.Simplify 3a + 7b - 4a + b.
Easy
- A-a + 8b
- B7a + 8b
- C-a + 6b
- D7a + 6b
2.Find the value of 7x + 3y when x = 12 and y = -6.
Easy
- A66
- B102
- C30
- D-18
3.Simplify 5c - d - 3d - 2c.
Easy
- A3c - 4d
- B7c - 4d
- C3c - 2d
- D7c - 2d
4.Complete the statement: When w = ______, 10w = 70.
Easy
- A7
- B700
- C0.7
- D60
5.Use the formula s = ut + 1/2 at². Calculate s when u = 5, t = 10 and a = 3.
Medium
- A200
- B80
- C350
- D65
6.Simplify 2p - q - 3q - 5p.
Easy
- A-3p - 4q
- B3p - 4q
- C-3p + 2q
- D7p - 4q
7.The body mass index B is given by B = m / h². Usman has mass 50 kg and height 1.57 m. Work out B correct to one decimal place.
Medium
- A20.3
- B31.8
- C20.2
- D32.0
8.Find the value of y when m = -2, x = -7 and c = -3 in y = mx + c.
Easy
- A11
- B-17
- C-11
- D17

Revision notes

## Algebra Toolkit - Night Before Exam Summary ### Algebraic Notation - Letters represent unknown numbers (variables). - Multiplication: no symbol, e.g., `ab` means `a × b`. - Division: use fractions, e.g., `a/b` means `a ÷ b`. - Powers: `a²` means `a × a`. - Brackets: `3(a + b)` means `3 × (a + b)`. ### Algebraic Vocabulary - **Term**: a number, variable, or product (e.g., `5x`, `-3y²`). - **Coefficient**: the number in front of a variable (e.g., in `6x`, coefficient is 6). - **Expression**: no equals sign (e.g., `2x + 5`). - **Equation**: has an equals sign, can be solved (e.g., `2x = 10`). - **Formula**: a rule relating quantities (e.g., `s = ut + ½at²`). ### Substitution - Replace letters with given numbers. - Use brackets around negative numbers: `(-3)`. - Follow order of operations (BIDMAS). - Example: If `x = 2`, `y = -4`, then `2x(x + 3y) = 2×2×(2 + 3×(-4)) = -40`. ### Collecting Like Terms - Like terms have same variables and powers (e.g., `2x` and `5x`). - Add/subtract coefficients only. - Example: `8a - 5b - 6a + 4b = 2a - b`. ### Key Formulas - `s = ut + ½at²` (displacement with constant acceleration). - `y = mx + c` (straight line). - `B = m/h²` (body mass index). ### Tips - Simplify `1x` to `x`. - For negative substitution, e.g., `x = -4`, `x² = (-4)² = 16`.

Variable: A letter representing an unknown or general number.
Coefficient: The number multiplying a variable (e.g., in 5x, coefficient is 5).
Term: A single number, variable, or product (e.g., 3x, -2y²).
Expression: An algebraic statement without an equals sign.
Equation: An algebraic statement with an equals sign that can be solved.
Formula: A rule or relationship between quantities using letters.
Substitution: Replacing a variable with a given number.
Like terms: Terms with exactly the same variables and powers.
Collecting like terms: Simplifying by adding/subtracting coefficients of like terms.

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