Introduction To Algebra
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Lesson notes
Algebraic Notation
- Algebra uses letters (variables) to represent general or unknown numbers.
- Algebraic notation writes calculations using letters: e.g., `a + b`, `ab` means `a × b`, `a/b` means `a ÷ b`.
- Multiplication is implied: `3ab` means `3 × a × b`.
- Order of operations applies: brackets, powers, then multiplication/division, then addition/subtraction.
- Powers and roots are written as with numbers: `a²` means `a × a`, `√a` means square root of `a`.
- Brackets work as with numbers: `3(a + b)` means `3 × (a + b)`.
Algebraic Vocabulary
- A term is a variable, a number (constant), or a product of numbers and variables (e.g., `5x`, `4xy`).
- The number in front of a variable is the coefficient (e.g., coefficient of `x` in `6x` is 6).
- A factor divides a term exactly; e.g., factors of `3x` are 1, 3, x, 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 linking quantities (e.g., `w = mg`); substituting values turns it into an equation.
Substitution
- Substitution replaces letters in a formula with given numbers.
- Always use brackets around negative numbers when substituting (e.g., `x = -3` → `x² = (-3)² = 9`).
- Follow order of operations: brackets, powers, multiplication/division, addition/subtraction.
- Substitution can produce an equation to solve for an unknown variable.
- Example: If `P = 2l + 2w`, `P = 20`, `w = 4`, then `20 = 2l + 8` → `l = 6`.
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`).
- To collect like terms, add or subtract their coefficients.
- Keep the sign with its term; `2x - 3y` is the same as `2x + (-3y)`.
- Simplify: `8a - 5b - 6a + 4b` → `(8a - 6a) + (-5b + 4b)` → `2a - b`.
- Do not leave `1x` or `-1x`; write `x` or `-x`.
Slides
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Practice questions
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1.Which of the following is an expression?
Easy- A2x + 5 = 4
- B7x - 9
- Cx = vt - w
- D3 + 4 = 7
2.Which of the following is an equation?
Easy- A2x + 5
- B7x - 9
- C2x + 5 = 4
- Dx = vt - w
3.Simplify: 6a - 3b + 2a - 4b
Medium- A8a - 7b
- B8a + 7b
- C4a - 7b
- D8a - b
4.Simplify: 4x + 3y + 2x - 8y
Medium- A6x - 5y
- B6x + 5y
- C2x - 5y
- D6x - 11y
5.W = 3a + 5c. Find the value of W when a = 6 and c = 2.
Easy- A28
- B23
- C48
- D36
6.Simplify: 3c - 5d - c + 2d
Medium- A2c - 3d
- B2c - 7d
- C4c - 3d
- D2c + 3d
7.Simplify: 5w + 3h - 7w + 8h
Medium- A-2w + 11h
- B2w + 11h
- C-2w - 5h
- D12w + 11h
8.Simplify: 5c - d - 3d - 2c
Medium- A3c - 4d
- B3c + 4d
- C7c - 4d
- D3c - 2d
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