Algebra - Solving Linear Equations in One Variable

A Linear Equation in one variable is an algebraic equation where the exponent of the variable is exactly $1$. Visually, if represented on a coordinate plane as $y = mx + c$, it would form a perfectly straight line.

The Equality Principle functions like a balance scale. To keep the equation true, any mathematical operation performed on the Left Hand Side (LHS) must also be performed on the Right Hand Side (RHS). Imagine adding weights to both sides of a scale to keep it level.

Isolation of the Variable is the primary goal of solving equations. This involves using inverse operations to move all constants to one side and the variable to the other, resulting in the form $x = \text{value}$.

Inverse Operations are pairs of mathematical operations that undo each other. Addition is the inverse of subtraction (e.g., to remove $-5$, add $+5$), and multiplication is the inverse of division (e.g., to remove a denominator of $3$, multiply by $3$).

The Distributive Property is used when an equation contains parentheses, such as $a(x + b) = c$. You must multiply the term outside the bracket by every term inside, visually expanding the expression to $ax + ab = c$ before solving.

Variables on Both Sides require grouping like terms. If you see $ax$ on the left and $cx$ on the right, subtract one of the variable terms from both sides so that the variable exists only on one side of the 'equals' sign.

Clearing Fractions is a technique used when equations involve denominators. By multiplying the entire equation by the Least Common Multiple (LCM) of all denominators, you can transform a fractional equation into a simpler linear form without fractions.