A practical guide to solving equations of all types — linear, quadratic, rational, and systems — with step-by-step methods and real-world applications.
What Is an Equation?
An equation is a mathematical statement that two expressions are equal. Solving an equation means finding the value(s) of the variable that make the statement true.
Types of Equations
Linear: ax + b = 0. Example: 3x + 7 = 22 → x = 5. Quadratic: ax² + bx + c = 0. Methods: factoring, quadratic formula, completing the square. Rational: Contains fractions with variables in denominator — multiply by LCD to clear fractions. Systems: Two or more equations with two or more variables. Methods: substitution, elimination, graphing.
Real-world example: Tutoring charges $50 registration + $35/hour. Budget $225: 35h + 50 = 225 → h = 5 hours.
Real-World Case Studies
Break-Even Analysis: Fixed costs $2,400, variable $4/mug, sell at $16. 16m = 2400 + 4m → m = 200 mugs/month. Projectile Motion: Ball thrown at 20 m/s: h = −4.9t² + 20t. Hits ground at t = 0 (start) or t ≈ 4.08 seconds.
Common Mistakes
- Forgetting to distribute negative signs: −(2x − 5) = −2x + 5
- Losing solutions when dividing by a variable
- Not checking for extraneous solutions in rational equations
- Misapplying PEMDAS when isolating — addition/subtraction first, then multiplication/division