Linear Equations in One Variable - Solving equations with variables on both sides

A Linear Equation in one variable is an algebraic equation where the highest power of the variable is 1. When solving equations with variables on both sides, the goal is to isolate the variable, typically represented as $x$, on one side of the equality sign ($=$).

The Balance Scale Concept: Think of an equation as a physical balance scale where the '=' sign is the fulcrum. To maintain equilibrium, any operation (addition, subtraction, multiplication, or division) performed on the left-hand side (LHS) must also be performed on the right-hand side (RHS). If you remove $3x$ from the left, you must remove $3x$ from the right to keep the 'beam' level.

Transposition Method: This involves moving terms from one side of the equation to the other. When a term moves across the '=' sign, its mathematical sign changes: addition becomes subtraction ($+ \to -$), subtraction becomes addition ($- \to +$), multiplication becomes division ($\times \to \div$), and division becomes multiplication ($\div \to \times$).

Grouping Like Terms: To solve efficiently, visualize all terms containing the variable moving to the left side and all constant numbers moving to the right side. This 'sorting' process allows you to combine coefficients into a single term like $ax = b$.

Using the Distributive Property: If the equation contains parentheses, such as $a(bx + c) = d$, visualize the multiplier '$a$' being 'distributed' or shared with every term inside the brackets. This step must be completed before you begin transposing terms across the equals sign.

Cross-Multiplication Visual: When an equation is in the form of two fractions set equal to each other, like \frac{ax+b}{p} = rac{cx+d}{q}, you can visualize an 'X' shape connecting the numerator of one side to the denominator of the other. Multiplying along these diagonal lines removes the fractions: $q(ax+b) = p(cx+d)$.

Verification: Always check your result by substituting the numerical value of the variable back into the original equation. If the LHS equals the RHS, your solution is correct, providing a visual confirmation of the balance.