The Gathering
What if you have \(x\)'s on both sides? To solve, you must "gather" all variables onto one side and all constants onto the other. It doesn't matter which side you choose, but usually, we aim for the side that keeps the variable positive.
Worked Examples
Example 1: Basic Gathering
Solve: \(5x - 3 = 2x + 9\)
- Step 1: Move the smaller \(x\) term. Subtract \(2x\) from both sides. \(3x - 3 = 9\).
- Step 2: Add 3 to both sides. \(3x = 12\).
- Step 3: Divide by 3. \(x = 4\).
- Result: \(x = 4\)
Example 2: Negative Variables
Solve: \(4 - 2x = 6x + 20\)
- Add \(2x\) to both sides. \(4 = 8x + 20\).
- Subtract 20 from both sides. \(-16 = 8x\).
- Divide by 8. \(x = -2\).
- Result: \(x = -2\)
Example 3: Distribution + Gathering
Solve: \(2(x + 4) = 5x - 1\)
- Expand: \(2x + 8 = 5x - 1\).
- Subtract \(2x\): \(8 = 3x - 1\).
- Add 1: \(9 = 3x\).
- Divide by 3: \(x = 3\).
- Result: \(x = 3\)
The Bridge to Quantum Mechanics
In Quantum Mechanics, the Schrödinger Equation often appears as: \[-\frac{\hbar^2}{2m} \psi'' + V\psi = E\psi\] Notice that the variable \(\psi\) is on both sides. To solve this, we gather all the \(\psi\) terms onto one side to find the "Characteristic Equation." This is the exact same logic as gathering \(x\)'s—we are consolidating our information so we can factor it and find the solution. Every step you take in basic algebra is a rehearsal for high-level physics.