Algebra Study Guide

Algebraic terminology and proofs are fundamental to higher-level mathematics. "Terminology" refers to the precise language and definitions used in algebra, such as 'identity', 'equation', 'expression', 'inequality', 'variable', 'coefficient', 'term', 'constant', 'polynomial', 'factor', 'multiple', 'rational number', 'irrational number', 'integer', 'consecutive integer', 'even number', 'odd number', and 'prime number'. Understanding these terms is crucial for interpreting mathematical problems and constructing correct solutions. For instance, an 'identity' is true for all values of its variables, while an 'equation' is only true for specific values. Recognising an 'expression' as something without an equals sign helps distinguish it from an equation.

"Proofs" in algebra involve demonstrating that a statement is true for all permissible values of a variable or variables, using logical steps and established mathematical facts. Unlike simply 'showing' something is true with an example, a proof must be general. Common proof techniques include direct proof, proof by exhaustion (checking all possible cases, often limited to a finite small set), and disproof by counterexample. For higher-tier GCSE, you will often encounter proofs involving sums, products, consecutive integers, or properties of even and odd numbers. For example, to prove that the sum of any two odd numbers is even, you would represent two arbitrary odd numbers as 2n+1 and 2m+1 (where n and m are integers), sum them to get 2n+1+2m+1 = 2n+2m+2 = 2(n+m+1). Since n+m+1 is an integer, 2(n+m+1) is an even number, thus proving the statement. It is vital to articulate each step clearly and to use correct algebraic notation throughout the proof. Students must move beyond merely verifying with specific numbers and instead establish general truths using algebraic manipulation and logical deduction.

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.

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.

A strong solution states the relevant rule, substitutes or transforms values carefully, shows intermediate reasoning, and checks that the result is reasonable and in the required form. Common errors come from omitted terms, mishandled signs, premature rounding or using a rule that does not match the structure of the problem.