Linear Equations in One Variable - Solving equations with linear expressions on one side and numbers on the other

A Linear Equation in one variable is an algebraic equation where the highest power of the variable is $1$. Visually, if this equation were plotted on a coordinate plane, it would form a straight line.

The Balance Scale Principle: An equation functions like a physical balance scale. To maintain equality, any operation (addition, subtraction, multiplication, or division) performed on the Left-Hand Side ($LHS$) must also be performed on the Right-Hand Side ($RHS$).

Transposition Method: This involves moving a term from one side of the equals sign to the other. When a term is transposed, its sign changes: $+b$ becomes $-b$, $-b$ becomes $+b$, multiplication becomes division, and division becomes multiplication. Imagine the '$=$' sign as a bridge that flips the operation.

Isolating the Variable: The objective is to rearrange the equation so the variable (like $x$) stands alone on one side. You 'undo' the operations surrounding the variable in reverse order of operations (usually handling addition/subtraction before multiplication/division).

Like Terms: Before solving, always simplify the linear expression by combining like terms. For example, $3x + 2x$ should be viewed as a single block of $5x$ to simplify the visual complexity of the equation.

Verification: Once a value for the variable is found, substitute it back into the original equation. If the $LHS$ simplifies to the same number as the $RHS$, the solution is correct.

The General Form: For this sub-topic, equations typically appear in the form $ax + b = c$, where $a, b,$ and $c$ are constants and $x$ is the variable. The goal is to move $b$ and then $a$ to solve for $x$.