Algebra Basics (AQA GCSE Maths): Revision Notes

Algebra basics

Before you can tackle more advanced algebraic concepts, you need to master some fundamental skills that form the foundation of all algebraic work. These basics will help you understand how to work with numbers, letters, and expressions confidently.

Working with negative numbers

Negative numbers appear constantly in algebra, so understanding how to handle them is essential for success. There are specific rules that govern how positive and negative numbers interact when you multiply or divide them.

These sign rules apply in two key situations:

When multiplying or dividing numbers: The pattern is straightforward - when the signs are the same, the result is positive; when the signs are different, the result is negative. For example, $-2 \times 3 = -6$ (different signs give negative), but $-8 \times -2 = +4$ (same signs give positive), and $-4p \times -2 = +8p$.

When two signs appear next to each other: This happens in expressions where you might see something like $5 - -4$. Here, the two minus signs together make a plus, so $5 - -4$ becomes $5 + 4 = 9$. Similarly, $x + -y$ becomes $x - y$, and $-z$ becomes $x - y + z$.

Understanding letters multiplied together

In algebra, letters are often written side by side without multiplication signs, which can initially seem confusing. Learning to interpret this notation correctly is crucial for working with algebraic expressions.

When letters appear next to each other, they represent multiplication:

• $abc$ means $a \times b \times c$ (the multiplication signs are omitted for simplicity)
• $gn^2$ means $g \times n \times n$ (only the $n$ is squared, not the $g$)
• $(gn)^2$ means $g \times g \times n \times n$ (the brackets show that both letters are squared)
• $p(q - r)^3$ means $p \times (q - r) \times (q - r) \times (q - r)$ (only the bracketed expression is cubed)

What are algebraic terms?

To work effectively with algebraic expressions, you need to understand their building blocks - terms. Recognising and identifying terms is fundamental to simplifying expressions and solving equations.

Each term has either a plus or minus sign in front of it. If you can't see a sign at the beginning of the first term, there's an invisible plus sign there.

Consider the expression $4xy + 5x^2 - 2y + 6y^2 + 4$:

• $4xy$ is an '$xy$' term
• $5x^2$ is an '$x^2$' term
• $-2y$ is a '$y$' term
• $6y^2$ is a '$y^2$' term
• $4$ is a 'number' term

Simplifying expressions by collecting like terms

One of the most important skills in algebra is simplifying expressions by combining similar terms. This process makes expressions more manageable and is essential for solving equations effectively.

The step-by-step process:

1. Identify each term by putting bubbles around them - Make sure you include the plus or minus sign that belongs to each term
2. Rearrange the terms into groups - Put like terms together so they're easier to combine
3. Combine the like terms - Add or subtract the numbers in front of the letters (called coefficients)

This process of collecting like terms is a fundamental skill that you'll use throughout your algebraic studies.