Multi-Step Algebraic Equations

When equations look messy because of fractions, we can use a 'magic' trick to simplify them: multiply every term by the Least Common Denominator (LCD). This 'clears' the fractions, leaving you with whole numbers that are much easier to manage. If you are dealing with decimals, you can do something similar by multiplying by powers of 10 (like 10, 100, or 1000) to shift the decimal point. Remember, the Golden Rule of Algebra still applies: whatever operation you perform on one side of the equation, you must perform on the other side to keep the scale balanced.

Sometimes, a variable is 'trapped' inside parentheses. To set it free, we use the Distributive Property: $a(b + c) = ab + ac$. When the number outside the parentheses is a fraction or a decimal, the process is the same. Be extra careful with negative signs! If you multiply a negative number across parentheses, it flips the sign of every term inside. Once the parentheses are gone, you can combine like terms and solve the equation using standard inverse operations (addition/subtraction, then multiplication/division).

A proportion is an equation stating that two ratios are equal, such as $\\frac{a}{b} = \\frac{c}{d}$. These are common in real-world scaling, like maps or recipes. To solve a complex proportion, we use cross-multiplication: $ad = bc$. This effectively clears both denominators at once! If the numerator or denominator contains an expression (like $x + 2$), treat it as a single unit by putting it in parentheses before you multiply. This ensures you distribute correctly.