Introduction To Algebra

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課程筆記

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`.

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練習題

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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

複習筆記

## Introduction to Algebra - Quick Revision ### 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 variable, constant, or product (e.g., `x`, `20`, `5x`). - **Coefficient**: the number in front of a variable (e.g., in `6x`, coefficient is 6). - **Expression**: no equals sign (e.g., `2x+5y`). - **Equation**: has an equals sign, can be solved (e.g., `2x=10`). - **Formula**: a rule linking quantities (e.g., `P=2l+2w`). ### Substitution - Replace letters with given numbers. - Use brackets around negative numbers: `x²` when x = -4 → `(-4)² = 16`. - Follow order of operations (BIDMAS). - Example: If `A = ¼ bc²`, b=3, c=6 → `A = ¼ × 3 × 36 = 27`. ### Collecting Like Terms - Like terms have same variable(s) and same powers. - Add/subtract coefficients only. - Example: `8a - 5b - 6a + 4b = (8a-6a) + (-5b+4b) = 2a - b`. - Simplify `1x` to `x`, `-1x` to `-x`. ### Key Formulas - Perimeter of rectangle: `P = 2l + 2w`. - Area of trapezium: `A = (a+b)h/2`. - From past paper: `c = 5a - 2b`; making a subject: `a = (c+2b)/5`.

Variable: A letter representing an unknown number.
Coefficient: The number multiplying a variable (e.g., in 5x, coefficient is 5).
Term: A single number, variable, or product of numbers and variables.
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 written in algebraic shorthand.
Substitution: Replacing a variable with a given numerical value.
Like terms: Terms with exactly the same variables and powers.
Collecting like terms: Simplifying an expression by adding/subtracting coefficients of like terms.

字卡

What is algebraic notation?

Algebraic notation uses letters to represent numbers. For example, 'ab' means a × b, and 'a/b' means a ÷ b.

How do you write multiplication in algebra?

No symbol is used; e.g., 'ab' means a × b, and '3ab' means 3 × a × b.

How do you write division in algebra?

Use fractions: 'a/b' means a ÷ b.

What does 'a²' mean?

a² means a × a (a squared).

What does '4a²' mean?

4a² means 4 × a × a (4 times a squared).

What does '√a' mean?

√a means the square root of a.

What does '3(a + b)' mean?

It means 3 × (a + b): add a and b first, then multiply by 3.

What is a term in algebra?

A term is a number, a variable, or a product of numbers and variables, e.g., 5x, 20, or x².

What is a coefficient?

The number in front of a variable; e.g., in 6x, the coefficient is 6; in -5y, it is -5.

What is a constant?

A constant is a term that does not contain a variable, e.g., 20.

What is an expression?

An algebraic statement without an equals sign, e.g., 2x + 5y.

What is an equation?

An algebraic statement with an equals sign that can be solved, e.g., 2x = 10.

What is a formula?

A rule or relationship written in letters, e.g., w = mg. It becomes an equation when values are substituted.

What does substitution mean?

Replacing a variable with a given number. Use brackets for negative numbers, e.g., if x = -4, then x² = (-4)² = 16.

Simplify: 6a - 3b + 2a - 4b

8a - 7b

Simplify: 4x + 3y + 2x - 8y

6x - 5y

Simplify: 3c - 5d - c + 2d

2c - 3d

Simplify: 5w + 3h - 7w + 8h

-2w + 11h

Simplify: 5c - d - 3d - 2c

3c - 4d

Simplify: 5t + 4t - 2t

Simplify: 8a + 3b - 2a + b

6a + 4b

Simplify: 4x - 12y + 10x + 25y

14x + 13y

Simplify: 2b + 3b - a - b + 3a

4b + 2a

Simplify: 9x - 2y - 5x - y

4x - 3y

If W = 3a + 5c, find W when a = 6 and c = 2.

W = 3(6) + 5(2) = 18 + 10 = 28

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