In one line
Gauss elimination uses row operations to turn the augmented matrix into upper-triangular form, then finds the unknowns by back-substitution from the last equation upwards.
Explained simply
Solving three equations at once feels hard. Gauss elimination makes it easy: remove \(x\) from equations 2 and 3, then remove \(y\) from equation 3. Now equation 3 has only \(z\), equation 2 has \(y\) and \(z\), and equation 1 has all three. Solve from the bottom up, like climbing down a ladder one rung at a time.
Technically, the augmented matrix \([A|B]\) is reduced by elementary row operations to \([U|C]\) with \(U\) upper triangular. The system \(UX=C\) is then solved by back-substitution. It is a direct method: it gives the exact answer (apart from rounding) in a fixed number of steps, unlike iterative methods. This is a self-learning topic.
How it works
- Write the augmented matrix.
- Using row 1 as the pivot row, make the entries below \(a_{11}\) zero.
- Using row 2, make the entry below \(a_{22}\) zero.
- Write the triangular system and solve for \(z\), then \(y\), then \(x\).
- If a pivot is zero, swap rows (partial pivoting also reduces rounding errors).
Key formula
Row operation: \(R_i\to R_i-\dfrac{a_{ik}}{a_{kk}}R_k\) to eliminate the entry \(a_{ik}\) below pivot \(a_{kk}\).
Quick example
Solve \(x+y+z=6\), \(2x+3y+4z=20\), \(3x+2y+5z=22\).
\(\left[\begin{array}{ccc|c}1&1&1&6\\2&3&4&20\\3&2&5&22\end{array}\right]\xrightarrow{R_2-2R_1,\ R_3-3R_1}\left[\begin{array}{ccc|c}1&1&1&6\\0&1&2&8\\0&-1&2&4\end{array}\right]\xrightarrow{R_3+R_2}\left[\begin{array}{ccc|c}1&1&1&6\\0&1&2&8\\0&0&4&12\end{array}\right]\)
Back-substitution: \(4z=12\Rightarrow z=3\); \(y+2(3)=8\Rightarrow y=2\); \(x+2+3=6\Rightarrow x=1\).
Check in equation 3: \(3+4+15=22\).
Common mistakes
- Changing only the left side of a row and forgetting the constant column.
- Back-substituting in the wrong order (starting from \(x\)).
- Dividing by a zero pivot instead of swapping rows.
How MU asks it
As a self-learning topic it may appear as a 5-mark part: "Solve by Gauss elimination method: ..." Show the augmented matrix, label each row operation, display the triangular form, and back-substitute clearly with a check in one original equation.