Introduction To Algebra

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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 $\times a \times$ 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 $\times (a +$ 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`.

Algebraic Notation Examples

Collecting Like Terms

Substitution with Negatives

Algebraic Vocabulary Overview

Practice questions

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1.Which of the following is an expression?
Easy
- A $2x + 5 = 4$
- B $7x - 9$
- C $x = vt - w$
- D $3 + 4 = 7$
2.Which of the following is an equation?
Easy
- A $2x + 5$
- B $7x - 9$
- C $2x + 5 = 4$
- D $x = vt - w$
3.Simplify: $6a - 3b + 2a - 4b$
Easy
- A $8a - 7b$
- B $8a + 7b$
- C $4a - 7b$
- D $8a - b$
4.Simplify: $4x + 3y + 2x - 8y$
Easy
- A $6x - 5y$
- B $6x + 5y$
- C $2x - 5y$
- D $6x - 11y$
5. $W = 3a + 5c$ . Find the value of W when $a = 6$ and $c = 2$ .
Easy
- A28
- B23
- C48
- D36
6.Simplify: $3c - 5d - c + 2d$
Easy
- A $2c - 3d$
- B $2c - 7d$
- C $4c - 3d$
- D $2c + 3d$
7.Simplify: $5w + 3h - 7w + 8h$
Easy
- A $-2w + 11h$
- B $2w + 11h$
- C $-2w - 5h$
- D $12w + 11h$
8.Simplify: $5c - d - 3d - 2c$
Easy
- A $3c - 4d$
- B $3c + 4d$
- C $7c - 4d$
- D $3c - 2d$

Summary

## 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`.

Key terms

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.

Slides

Algebraic Notation

Algebra uses letters to represent unknown numbers; these are called variables.
Multiplication is written without a symbol: ab means $a \times b$ .
Division is written as a fraction: a/b means $a \div b$ .
Powers and brackets work the same as with numbers.

What does 3ab mean?

$3 + a + b$
$3 \times a \times b$
$a + b + 3$
$3a + b$

Algebraic Vocabulary

A term is a number, variable, or product of numbers and variables (e.g., 5x, 20).
The number in front of a variable is called the coefficient (e.g., in 6x, the coefficient is 6).
An expression has no equals sign; an equation has an equals sign and can be solved.
A formula shows a relationship between quantities $(e.g., w = mg)$ .

In the term 7y, the coefficient $is ____$ . $An ____$ does not have an equals sign.

Substitution

Substitution means replacing a variable with a given number.
Always use brackets around negative numbers when substituting (e.g., (-3)).
Follow the order of operations (BIDMAS) when calculating the value.

If $x = 3$ and $y = -2$ , what is the value of $4x^{2} + xy$ ?

Collecting Like Terms

Like terms have exactly the same variables and powers (e.g., 2x and 3x).
To simplify, add or subtract the coefficients of like terms.
The sign in front of a term moves with it when rearranging.

Which of these are like terms?

3x and 5x
4y and $4y^{2}$
2xy and 3yx
a and b

Simplifying Expressions

Collect like terms separately for each variable.
For example, $8a - 5b - 6a + 4b = (8a - 6a) + (-5b + 4b) = 2a - b$ .
Do not leave coefficients of 1 or -1; write x instead of 1x.

Simplify: $6a - 3b + 2a - 4b = ____ a - ____ b$

Substitution into Formulas

Substitute given values into a formula to find an unknown.
Sometimes this creates an equation that you can solve.
Example: $P = 2l +$ 2w; if $P = 20$ and $w = 4$ , then $20 = 2l + 8$ , so $l = 6$ .

Given $A =$ ¼ $b c^{2}$ , find A when $b = 3$ and $c = 6$ .

Flashcards

What is algebraic notation?

Algebraic notation uses letters to represent numbers. For example, 'ab' means $a \times b$ , and 'a/b' means $a \div b$ .

How do you write multiplication in algebra?

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

How do you write division in algebra?

Use fractions: 'a/b' means $a \div b$ .

What does $'a^{2}'$ mean?

$a^{2}$ means $a \times a (a squared)$ .

What does $'4a^{2}'$ mean?

$4a^{2}$ means $4 \times a \times a (4$ times a squared).

What does $'\sqrt{a}'$ mean?

$\sqrt{a}$ means the square root of a.

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

It means $3 \times (a + b)$ : 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 $x^{2}$ .

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 $x^{2} = (-4)^{2} = 16$ .

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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