Complete Algebra Booster Lecture 1: Equations

Vocabulary flashcards covering core algebraic concepts, equation-solving techniques, and system of linear equation conditions from Lecture 1.

Elimination Method

A technique for solving a system of linear equations by adding or subtracting equations to eliminate one variable, making it possible to solve for the remaining variable.

Substitution Method

A technique for solving a system of linear equations by expressing one variable in terms of the other from one equation and substituting that expression into the second equation.

System of Linear Equations Cheat Sheet

A reference cheat sheet outlining the three possible outcomes for a system of linear equations $a_1 x + b_1 y + c_1 = 0$ and $a_2 x + b_2 y + c_2 = 0$ based on coefficient ratios.

Unique Solution Condition

The algebraic condition $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$ for two linear equations, indicating that their lines intersect at exactly one point and the system is consistent and independent.

No Solution Condition

The algebraic condition $\frac{a_1}{a_2} = \frac{b_1}{b_2} \neq \frac{c_1}{c_2}$ for two linear equations, indicating that the lines are parallel, never meet, and the system is inconsistent.

Infinite Solutions Condition

The algebraic condition $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ for two linear equations, indicating that the lines overlap entirely and the system is consistent and dependent.

CAT Hack for Unique Solutions

A shortcut rule stating that if $\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$, a unique solution is instantly confirmed without needing to check $\frac{c_1}{c_2}$.

Quadratic Formula

The standard formula used to find roots for any quadratic equation $a x^2 + b x + c = 0$, given by $x = \frac{-b \pm \sqrt{b^2 - 4 a c}}{2 a}$.

Sum of Perfect Squares Principle

An observation method where an equation written as a sum of non-negative squared terms equal to zero, such as $(x - 3)^2 + y^2 = 0$, requires each squared term to independently equal zero for real numbers.

Algebraic Optimisation

A problem-solving strategy focused on determining the maximum or minimum values of algebraic expressions under specific integer or real constraints.

Parallel Lines Condition in General Form

For two lines $a x + b y = 6$ and $7 x + 4 y = 8$ to be parallel, their variable coefficient ratios must be equal, meaning $\frac{a}{7} = \frac{b}{4}$.