Algebra
What the domain tests
Algebra in the basic-school exam means manipulating expressions, solving equations and inequalities, handling linear systems, and rearranging formulas so that a named variable becomes the subject. Questions often come as a mixture: simplify first, then solve, then check whether the solution is allowed.
Core rules
- Expanding: a(b + c) = ab + ac; (a + b)² = a² + 2ab + b²; (a − b)² = a² − 2ab + b²; (a + b)(a − b) = a² − b².
- Factorising: take the common factor out first, then look for a difference of squares, then for a quadratic of the form x² + (p + q)x + pq = (x + p)(x + q).
- Linear equations: perform the same operation on both sides; clear fractions by multiplying every term by the lowest common denominator; finish by substituting back to check.
- Inequalities: the solution set is an interval, and multiplying or dividing both sides by a negative number reverses the inequality sign.
- Systems of two linear equations: either substitution (express one variable from the simpler equation) or addition/subtraction after matching coefficients. One solution, no solution (parallel lines) or infinitely many (same line) are all possible.
- Quadratic equations: rearrange to ax² + bx + c = 0, then factorise, or use x = (−b ± √(b² − 4ac)) / (2a). Vieta: the roots sum to −b/a and multiply to c/a.
- Formulas: undo operations in reverse order and treat the wanted letter as the unknown.
- Domain conditions: any denominator must not be zero.
Worked example
Solve the system 2x + 3y = 12 and x − y = 1. From the second equation x = y + 1. Substitute: 2(y + 1) + 3y = 12, so 2y + 2 + 3y = 12, 5y = 10, y = 2. Then x = 2 + 1 = 3. Check in the first equation: 2·3 + 3·2 = 6 + 6 = 12, correct.
Traps that cost marks
- Sign errors when a minus sign stands before a bracket: −(a − b) = −a + b.
- Forgetting to reverse an inequality after multiplying by a negative number.
- Dividing both sides by x and losing the solution x = 0.
- Accepting a solution that makes an original denominator zero.
- Giving a bare number when the question asks for both variables, or for the solution set as an interval.