Algebra Revision Notes

Understanding Core Algebraic Terminology

Algebra uses special words to describe its components. An 'expression' is a combination of numbers, variables, and arithmetic operations, such as 2x + 7. It does not contain an equals sign. An 'equation' is a statement that two expressions are equal, for example, 2x + 7 = 15. A 'formula' is a specific type of equation that shows the relationship between variables, often used to calculate a quantity, like V = IR (Ohm's Law). An 'identity' is an equation that is true for all possible values of its variables; for example, (a+b)^2 = a^2 + 2ab + b^2 is always true. A 'term' is a single part of an expression separated by addition or subtraction, like 5x or 8. The 'coefficient' is the number multiplying a variable (e.g., 5 in 5x), and a 'constant' is a term without a variable (e.g., 8).

Constructing Simple Algebraic Proofs

Algebraic proofs demonstrate that a mathematical statement is true. At GCSE level, this often involves showing an identity means both sides of an equation are equal, or proving properties of numbers (like odd/even). To prove an identity, you typically expand and simplify one or both sides until they are identical. For example, to prove x(x+1) + x(x-1) = 2x^2, you expand the left side: x^2 + x + x^2 - x = 2x^2. Since 2x^2 is equal to the right side, the identity is proven. Remember to show each logical step clearly.

Proving Properties of Odd and Even Numbers

To prove properties involving odd and even numbers, represent them algebraically. An 'even' number can be represented as 2n, where n is an integer. An 'odd' number can be represented as 2n+1 (or 2n-1), where n is an integer. For example, to prove that the sum of two even numbers is always even: Let the two even numbers be 2m and 2n. Their sum is 2m + 2n = 2(m+n). Since (m+n) is an integer, 2(m+n) is a multiple of 2, and therefore an even number. Always state your representation clearly (e.g., 'Let n be an integer').

Key Algebraic Terminology

Understanding basic terms is vital. A 'variable' is a letter representing an unknown value (e.g., x, y). A 'constant' is a fixed numerical value. A 'term' is a part of an expression separated by + or - signs (e.g., in 2x + 3, '2x' and '3' are terms). A 'coefficient' is the number multiplying a variable in a term (e.g., 2 in 2x). An 'expression' is a combination of terms, variables, and numbers (e.g., 5x + 7). An 'equation' uses an equals sign to show two expressions are equivalent (e.g., 5x + 7 = 17). An 'identity' is an equation true for all values of its variables, often indicated by three horizontal lines (≡).

Simplifying Expressions

Simplifying expressions involves collecting 'like terms'. Like terms have the same variables raised to the same powers. For example, 3x and 5x are like terms, but 3x and 3x² are not. To collect like terms, add or subtract their coefficients. Example: 3x + 5 + 2x - 1 = (3x + 2x) + (5 - 1) = 5x + 4. Be careful with signs when subtracting. Simplifying helps make expressions clearer and easier to work with, which is often a first step in solving equations or proving identities.

Basic Algebraic Proofs

Algebraic proofs at foundation level often involve demonstrating that an identity is true. This typically means starting with one side of an equation (usually the more complex side) and manipulating it using algebraic rules (like distribution, combining like terms) until it matches the other side. For example, to prove x(x+y) - y(x+2) ≡ x² + xy - yx - 2y, you would expand the brackets on the left side to get x² + xy - yx - 2y. Since this matches the right side, the identity is proven. Always show each step clearly and provide reasons if necessary.

Common Proof Techniques and Understanding 'Always True'

In algebraic proofs, you might be asked to show that a statement is 'always true' for any value of a particular variable. This often involves rearranging or simplifying an expression to show that it consistently holds. For example, proving that 'the sum of two consecutive integers is always odd' can be done by letting the first integer be 'n'. The next consecutive integer is 'n+1'. Their sum is n + (n+1) = 2n + 1. Since 2n is always an even number, 2n + 1 is always an odd number. This demonstrates the statement is always true. This type of reasoning relies on definition and algebraic manipulation.

Key Algebraic Terminology for Proofs

Mastering the precise definitions of algebraic terms is essential for constructing and understanding proofs. An 'identity' is an equation that holds true for all possible values of its variables, symbolised often with $\equiv$, e.g., $(x+y)^2 \equiv x^2 + 2xy + y^2$. An 'equation' holds true for specific values. An 'expression' is a combination of terms without an equals sign. You must distinguish between 'integers' (whole numbers, positive, negative or zero), 'rational numbers' (can be written as a fraction $p/q$ where $p, q$ are integers and $q \ne 0$), and 'irrational numbers' (cannot be written as such a fraction, e.g., $\sqrt{2}$). 'Consecutive integers' follow each other, like $n, n+1, n+2$. 'Even numbers' are multiples of 2, represented as $2n$. 'Odd numbers' are represented as $2n+1$ or $2n-1$. A 'prime number' has exactly two distinct positive divisors: 1 and itself.

Strategies for Algebraic Proofs

The goal of an algebraic proof is to demonstrate a statement is true for *all* relevant cases, not just specific examples. Direct proof is the most common method: start with known facts or definitions and use logical steps and algebraic manipulation to arrive at the desired conclusion. Proof by exhaustion is applicable when there are a finite and small number of cases to check; every case must be verified. Disproof by counterexample involves finding just one specific case where the statement is false. When proving statements involving sums or products, it's often useful to represent general forms of numbers, e.g., two consecutive integers could be $n$ and $n+1$, two even numbers could be $2a$ and $2b$, and two odd numbers could be $2a+1$ and $2b+1$. Always ensure your algebraic steps are clear, accurate, and logically flow from one line to the next. The conclusion of the proof should explicitly state that the property has been shown to be true for all cases.

Core method

Like terms have exactly the same variable part, including the same powers. Collect them by adding or subtracting their coefficients. Constants collect with constants, but unlike terms must remain separate. The operation changes the number of terms, not the variables themselves. The essential approach is to identify the mathematical structure before calculating, retain exact values where possible, and show enough working for each conclusion to be checked independently.

Worked reasoning and checks

For example, 4x + 7 + 3x − 2 = 7x + 5. In 5a + 2b − 3a + 6b, collect the a-terms and b-terms separately to obtain 2a + 8b. Terms such as x and x² are unlike because their powers differ. Check the final result against the original information, including signs, units, scale and whether the requested form or degree of accuracy has been supplied.