The Core of Algebra
A Variable is a letter (like \(x, y, t\) or \(\psi\)) that stands in for a number. Variables allow us to write "General Truths" that apply to any situation.
Expressions vs. Equations
- Expression: A mathematical "phrase" like \(2x + 5\). You can't "solve" it, but you can Evaluate it if you know what \(x\) is.
- Equation: A mathematical "sentence" with an equals sign, like \(2x + 5 = 11\). You can Solve this to find \(x\).
Worked Examples
Example 1: Evaluating an Expression
Evaluate \(3x - 4\) when \(x = 5\).
- Replace \(x\) with 5: \(3(5) - 4\).
- Follow PEMDAS: \(15 - 4 = 11\).
- Result: 11
Example 2: Negative Substitution
Evaluate \(x^2 + 2x\) when \(x = -3\).
- Replace \(x\) with (-3): \((-3)^2 + 2(-3)\).
- \((-3)^2\) is 9. \(2(-3)\) is -6.
- Combine: \(9 - 6 = 3\).
- Result: 3
Example 3: Multiple Variables
Evaluate \(\frac{x+y}{z}\) when \(x=10, y=2, z=4\).
- Replace: \(\frac{10+2}{4}\).
- Simplify top: \(\frac{12}{4} = 3\).
- Result: 3
The Bridge to Quantum Mechanics
In Quantum Mechanics, we use the Greek letter Psi (\(\psi\)) as our primary variable. It doesn't represent a single number, but a whole set of information called a "State." Just as you substituted \(x=5\) into an expression, physicists substitute the properties of a specific atom into the variable \(\psi\) to calculate its energy. Learning to treat letters as numbers is the single most important skill for a quantum architect.