Gaussian elimination performed in finite-precision arithmetic can become inaccurate when it divides by a very small pivot.

Partial pivoting swaps rows so that the pivot is chosen from a larger-magnitude entry in the current column before elimination proceeds.

For example, replacing a tiny pivot $a_{kk}$ by a larger available entry reduces the size of multipliers such as

$$m_{ik}=\frac{a_{ik}}{a_{kk}},$$

which helps limit amplification of rounding errors.

Pivoting does not change the linear system's solution; it changes the numerical path used to reach it.

Stable linear-system solution therefore requires more than the algebraic fact that row operations preserve solutions. The ordering of those operations and the magnitudes encountered during elimination matter in floating-point arithmetic.