Geometric Bites

Essential Elements of Algebra Problem Solving Solving equations in algebra is about preserving truth while working step by step toward the unknown. Although equations can look complicated, most of the time we are simply applying a small set of rules with care. If these rules are followed, the solution you reach will be valid; if they are broken, the result becomes unreliable. This post introduces three essential principles that underpin almost all algebraic manipulation. Follow them consistently, and you will have a strong foundation for solving equations with confidence. 1) Start with a True Statement Everything in algebra begins with a statement that is already true. If the starting point is false, no amount of manipulation can produce a trustworthy conclusion. When you write an equation such as: 2x + 3 = 13 you are asserting that doubling a number and then adding three gives thirteen. This is your initial truth. All further steps must preserve this truth. If y...

Function Composition: A Simple Way to Organise Your Mathematics Function composition is one of the most useful tools in mathematics. It allows us to combine several steps into a single process, keeping our work neat, organised, and easy to reuse. Rather than performing one operation after another by hand, composition lets us build those steps into a single function. Once you become comfortable with function composition, you never want to go back to doing everything manually. It reduces clutter, helps you work systematically, and allows you to achieve remarkable results with only a few lines of equations. What Is Function Composition? A function takes an input, performs an operation, and produces an output. Function composition takes this idea further: it links functions together so that the output of one becomes the input of the next. We write this as: (f ∘ g)(x) = f(g(x)) This means that g acts first, then f . The circle symbol ∘ simply means “do one funct...