Solving Systems of Linear Equations: Elimination
Elimination Method
Elimination reduces a system to a simpler equivalent system by applying three legal row operations:
- Swap two equations
- Multiply an equation by a non-zero scalar
- Add a multiple of one equation to another
These operations do not change the solution set.
Example: Forward Elimination
\[\begin{cases} 2x + y - z = 8 \\ -3x - y + 2z = -11 \\ -2x + y + 2z = -3 \end{cases}\]
Step 1 — Eliminate \(x\) from equations 2 and 3 using equation 1 as pivot:
\(R_2 \leftarrow R_2 + \frac{3}{2}R_1\):
\[-3x - y + 2z + \frac{3}{2}(2x + y - z) = -11 + \frac{3}{2}(8) \Rightarrow \frac{1}{2}y + \frac{1}{2}z = 1\]
\(R_3 \leftarrow R_3 + R_1\):
\[-2x + y + 2z + (2x + y - z) = -3 + 8 \Rightarrow 2y + z = 5\]
Step 2 — Eliminate \(y\) from equation 3:
\(R_3 \leftarrow R_3 - 4R_2\): \(\Rightarrow -z = 1 \Rightarrow z = -1\)
Back substitution: \(y = 2\), \(x = 2\).
Augmented Matrix Notation
The same process on the augmented matrix \([A|\mathbf{b}]\):
\[\left[\begin{array}{ccc|c} 2 & 1 & -1 & 8 \\ -3 & -1 & 2 & -11 \\ -2 & 1 & 2 & -3 \end{array}\right]\]