Algebra Toolkit

Algebraic Notation

• Algebra uses letters to represent unknown numbers, 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: $a^{2}$ means $a \times a, 3(a+b)$ means $3 \times (a+b)$.

What does 4xy mean?

• $4 + x + y$
• $4 \times x \times y$
• $4x + y$
• $x + y + 4$

Algebraic Vocabulary

• A term is a number, variable, or product of numbers and variables $(e.g., 5x, -3y^{2})$.
• The number in front of a variable is the coefficient (e.g., in 6x, the coefficient is 6).
• An expression has no equals sign $(e.g., 2x + 3y)$.
• An equation has an equals sign and can be solved $(e.g., 2x = 10)$.
• A formula shows a relationship between quantities $(e.g., P = 2l + 2w)$.

Which of these are true?

• An expression contains an equals sign.
• In 5x, the coefficient of x is 5.
• A formula can be solved without extra information.
• A constant is a term that cannot change value.

Substitution

• Substitution means replacing a variable with a given number.
• Always put brackets around negative numbers when substituting (e.g., (-3)).
• Follow the order of operations (BIDMAS) when simplifying.
• If you get an equation after substitution, you may need to solve it.

If $x = -4$, then x^{2} = ____.

Collecting Like Terms

• Like terms have the same variable(s) and the same powers (e.g., 2x and 5x).
• Unlike terms have different variables or powers (e.g., 2x and 2y).
• To simplify, add or subtract the coefficients of like terms.
• The sign in front of a term belongs to that term (e.g., -3y has coefficient -3).

Simplify $3a + 7b - 4a + b$.

• $-a + 8b$
• $7a + 8b$
• $-a + 6b$
• $a + 8b$

More on Collecting Like Terms

• Remember that xy and yx are like terms (multiplication is commutative).
• Be careful with powers: $x^{2}$ and x are not like terms.
• Always simplify coefficients: 1x becomes x, -1x becomes -x.
• Work step by step: group like terms, then combine.

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

Substitution with Formulas

• Formulas like $s = ut +$ ½at² are common in IGCSE.
• Substitute given values carefully, especially negatives.
• Use brackets for negative numbers to avoid sign errors.
• Sometimes you need to solve for a variable after substitution.

Using $s = ut +$ ½at², find s when $u=5, t=10, a=3$.

• 200
• 50
• 150
• 250

Algebraic Notation: Worked Example

• Raheem starts with q points and earns 6 more: total $= q + 6$.
• If his total is then doubled: $2(q + 6)$ or $2q + 12$.
• Brackets are used to show the doubling applies to the whole sum.

Which of these are correct algebraic expressions?

• $3(a + b)$ means $3 \times (a + b)$
• ab means $a + b$
• a/b means $a \div b$
• $4a^{2}$ means $4 \times a \times a$

Substitution with Negative Numbers

• When substituting negatives, use brackets: $e.g., x = -4$, then $x^{2} = (-4)^{2} = 16$.
• Be careful with powers: $(-4)^{2} = 16$ $but -4^{2} = -16$.
• For expressions like $y^{2} + y^{3}$ with $y = -4: (-4)^{2} + (-4)^{3} = 16 + (-64) = -48$.

If $y = -2$, find w = 5y^{2} - y^{3}: ____.

Collecting Like Terms: Worked Example

• Simplify $8a - 5b - 6a + 4b$.
• Collect a terms: $8a - 6a = 2a$.
• Collect b terms: $-5b + 4b = -b$.
• Answer: $2a - b$.

Simplify $2p - q - 3q - 5p$.

• $-3p - 4q$
• $7p - 4q$
• $-3p + 2q$
• $3p - 4q$