Algebra, Equations and Matrices
This topic tests symbolic manipulation: expressions, linear equations, inequalities, identities, and basic matrices.
- Expressions and substitution:
- An algebraic expression contains variables, constants, and operations (e.g., 3x + 2y - 5).
- Substitution means replacing variables with given numeric values to evaluate the expression.
- Linear equations:
- Solved by combining like terms and keeping both sides balanced — any operation done to one side must be done to the other.
- Transposition rule: When a term moves across the equals sign, its sign changes (e.g., 2x + 3 = 9 → 2x = 9 - 3 = 6 → x = 3).
- Inequalities:
- Solved the same way as equations, except: multiplying or dividing both sides by a negative number reverses the inequality sign.
- Example: -2x > 6 → dividing both sides by -2 gives x < -3 (sign flipped).
- Identities: Equations that are true for all values of the variable, not just specific ones. Key identity to memorize:
- Difference of squares: a^2 - b^2 = (a - b)(a + b). Useful for quick factoring and mental-math tricks (e.g., 99 x 101 = (100-1)(100+1) = 100^2 - 1 = 9999).
- Matrices:
- A matrix is a rectangular array of numbers arranged in rows and columns.
- Size is written as rows x columns — a 2x3 matrix has 2 rows and 3 columns.
- Basic operations: matrices of the same size can be added/subtracted by combining corresponding elements; matrix multiplication requires the number of columns of the first to match the number of rows of the second.
Test tip: For inequality questions, always double-check whether you multiplied/divided by a negative number — forgetting to flip the sign is the single most common error.