Algebra
Algebra tests symbolic fluency: simplifying expressions, solving linear equations and inequalities, systems, exponent and polynomial operations, factoring, and quadratic equations. The exam favors multi-step problems where one sign slip changes the answer.
Core rules
- Distribute carefully: a − (b + c) = a − b − c. Combine only like terms.
- Solve linear equations by applying inverse operations to both sides; keep the equation balanced.
- For inequalities, reverse the inequality symbol when multiplying or dividing both sides by a negative number.
- Systems: substitution isolates one variable; elimination adds or subtracts multiples of equations to cancel a variable.
- Polynomials: multiply each term of one factor by each term of the other, then combine like terms.
- Factor by greatest common factor first, then difference of squares a² − b² = (a − b)(a + b), then trinomials.
- Quadratic formula: x = [−b ± √(b² − 4ac)]/(2a). The discriminant b² − 4ac tells how many real solutions exist.
Worked example
Solve the system 2x + 3y = 12 and x − y = 1. From the second equation, x = y + 1. Substitute into the first: 2(y + 1) + 3y = 12, so 2y + 2 + 3y = 12, 5y = 10, y = 2. Then x = 2 + 1 = 3. Check: 2(3) + 3(2) = 12 and 3 − 2 = 1. The solution is (3, 2).
Traps that cost marks
- Dropping the negative when distributing, especially before parentheses.
- Forgetting to flip an inequality after dividing by a negative.
- Dividing both sides by a variable and losing the solution where that variable equals zero.
- Adding exponents when multiplying terms with different bases, or distributing an exponent over a sum: (x + y)² is not x² + y².
- Accepting a solution without checking it in the original equation, which catches sign errors and extraneous roots.