Solving in 3D
A system with three variables (\(x, y, z\)) represents three planes in 3D space. The solution is the single point where all three planes intersect. To solve this, we use elimination twice to reduce it to a 2-variable system.
Worked Example
Solving a 3x3 System
Solve: \[x + y + z = 6 \quad (1)\] \[2x - y + z = 3 \quad (2)\] \[x + 2y - z = 2 \quad (3)\]
- Step 1: Add (1) and (2) to eliminate \(y\). \(3x + 2z = 9 \quad (A)\).
- Step 2: Multiply (2) by 2 and add to (3) to eliminate \(y\). \(4x - 2y + 2z = 6\) added to \(x + 2y - z = 2\) gives \(5x + z = 8 \quad (B)\).
- Step 3: Solve the system (A) and (B). From (B), \(z = 8 - 5x\).
- Substitute into (A): \(3x + 2(8 - 5x) = 9 \to 3x + 16 - 10x = 9 \to -7x = -7 \to x = 1\).
- Find \(z\): \(z = 8 - 5(1) = 3\).
- Find \(y\): Substitute \(x=1, z=3\) into (1). \(1 + y + 3 = 6 \to y = 2\).
- Result: \((1, 2, 3)\)
The Bridge to Quantum Mechanics
The universe is 3-dimensional. Every quantum particle has three position variables (\(x, y, z\)) and three momentum variables (\(p_x, p_y, p_z\)). When we calculate the "Probability Cloud" of an electron in a hydrogen atom, we are solving systems of equations in three dimensions simultaneously. This lesson is your first step into "Vector Math"—the only math capable of describing our physical world.