Algebra - Solving Equations and Word Problems

A Linear Equation in one variable is a mathematical statement where two expressions are equal, and the highest power of the variable (usually $x$, $y$, or $z$) is $1$. Visually, this can be represented as a balance scale where the left-hand side (LHS) and right-hand side (RHS) must stay level.

The Principle of Equality states that you can add, subtract, multiply, or divide (by a non-zero number) the same value on both sides of the equation without changing the equality. Imagine adding or removing equal weights from both pans of a balance scale; the scale remains perfectly horizontal.

Transposition is the process of moving a term from one side of the equation to the other. When a term moves across the equal sign ($=$), its operation changes: $+$ becomes $-$, $-$ becomes $+$, $\times$ becomes $\div$, and $\div$ becomes $\times$. Visually, think of the equal sign as a 'magic gate' that flips the sign of any term passing through it.

Solving Equations with Brackets involves using the Distributive Property, where a term outside the parentheses is multiplied by every term inside, such as $a(b + c) = ab + ac$. Visually, this is like distributing a gift to every person inside a room.

Cross-Multiplication is used when the equation is in the form of a proportion, like $\frac{a}{b} = \frac{c}{d}$. We multiply the numerator of the first side by the denominator of the second side and vice-versa, resulting in $ad = bc$. Visually, draw an 'X' shape connecting the top of one side to the bottom of the other.

Translating Word Problems involves identifying keywords: 'sum' or 'increased by' means $+$, 'difference' or 'diminished by' means $-$, 'product' or 'times' means $\times$, and 'is' or 'results in' means $=$. Visualizing the problem as a flow diagram helps in constructing the algebraic equation correctly.