Algebraic Foundations & Equations

TL;DR

Algebra uses letters (variables) to represent unknown numbers, allowing you to solve problems by finding those unknowns. Equations show that two expressions are equal, and you can manipulate them to isolate the variable. Think of solving equations as balancing a scale: whatever you do to one side, you must do to the other to keep it balanced.

1. The Mental Model

Imagine an algebraic equation like a balanced scale. Each side has some items, and the goal is to figure out the weight of a mystery item (your variable) while keeping the scale perfectly balanced. Whatever you add or remove from one side, you must do the exact same to the other.

2. The Core Material

When we talk about algebraic foundations, we're mostly talking about understanding variables, expressions, and how to solve equations.

What are Variables and Expressions?

A variable is simply a letter, like x, y, or a, that stands in for an unknown number. We use variables when we want to talk about "some number" without knowing exactly what it is yet, or when a number can change.

An expression is a combination of numbers, variables, and operation signs (+, -, , /). It doesn't have an equals sign.
* Examples of expressions:* x + 5, 2y - 7, 3a/4, x^2

What is an Equation?

An equation is a statement that two expressions are equal. It always has an equals sign (=). The goal is often to find the value(s) of the variable(s) that make the equation true.
* Examples of equations: x + 5 = 10, 2y - 7 = 3, 3a/4 = 6

Solving Basic Equations

The main idea for solving equations is to isolate the variable. This means getting the variable all by itself on one side of the equals sign. To do this, you'll perform inverse operations.

• Addition and Subtraction are inverse operations:

  • If you have x + 3 = 7, to get x alone, you subtract 3 from both sides:
    x + 3 - 3 = 7 - 3
x = 4
  • If you have y - 5 = 2, to get y alone, you add 5 to both sides:
    y - 5 + 5 = 2 + 5
y = 7

• Multiplication and Division are inverse operations:

  • If you have 2x = 8, to get x alone, you divide both sides by 2:
    2x / 2 = 8 / 2
x = 4
    (Remember, 2x means 2 * x)
  • If you have x / 3 = 5, to get x alone, you multiply both sides by 3:
    (x / 3) * 3 = 5 * 3
x = 15

Combining Like Terms

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