Adding to Erase
The Elimination Method (also called Addition/Subtraction) involves adding the two equations together to make one variable disappear. This is usually faster than substitution when both equations are in standard form (\(Ax + By = C\)).
Worked Examples
Example 1: Direct Elimination
Solve: \[x + y = 10\] \[x - y = 4\]
- Add the equations: \((x + x) + (y - y) = 10 + 4\).
- \(2x = 14 \to x = 7\).
- Substitute back: \(7 + y = 10 \to y = 3\).
- Result: \((7, 3)\)
Example 2: Multiplication First
Solve: \[2x + 3y = 12\] \[x - y = 1\]
- Multiply the second equation by 3: \(3x - 3y = 3\).
- Now add to the first: \((2x + 3x) + (3y - 3y) = 12 + 3\).
- \(5x = 15 \to x = 3\).
- Find \(y\): \(3 - y = 1 \to y = 2\).
- Result: \((3, 2)\)
The Bridge to Quantum Mechanics
When we work with Matrices in Chapter 9, "Elimination" becomes the primary tool for solving any physical problem. This technique is known as Gaussian Elimination. In Quantum Mechanics, we use it to find the energy levels (eigenvalues) of complex molecules. By "eliminating" variables, we are effectively simplifying the coordinate system until the physical answer reveals itself. It is the core algorithm behind almost all physics simulation software.