Gauss elimination with pivoting is an efficient method for solving systems of linear equations. It involves transforming the system into an upper triangular form by eliminating variables one by one. Pivoting is used to prevent division by zero and to reduce errors caused by subtractive cancellation.

Here’s how the Gauss elimination with pivoting works:

1. Pivoting: Before starting the elimination process, rearrange the equations to place the equation with the largest coefficient of the variable corresponding to the first column at the top. This is done to reduce the effects of rounding errors during division.
2. Elimination: Starting with the first equation, use it to eliminate the first variable from all subsequent equations. Then proceed to the second equation to eliminate the second variable from subsequent equations, and so on.
3. Back Substitution: Once the system is in upper triangular form, solve for the variables starting from the last equation and substituting the known values back into the previous equations.

Here’s an example to illustrate the method:

Consider the system of equations:

1. Pivoting: Rearrange the equations to place the equation with the largest coefficient of the variable corresponding to the first column at the top. In this case, the first equation already has the largest coefficient for $x$, so no rearrangement is needed.
2. Elimination:Start with the first equation: 2x+y−z=8

   Multiply the first equation by the appropriate scalar values and subtract them from the subsequent equations to eliminate the x term:

   2x+y−z=8−23​(2x+y−z)+2z=−11−22​(2x+y−z)+2z=−3​

   This simplifies to:

   ​2x+y−z=8−25​y+21​z=−5−3y+4z=2​

3. Back Substitution:Solve the third equation for y=34z−2​

   Substitute this expression into the second equation and solve for  z:−52(4z−23)+12z=−5 −25​(34z−2​)+21​z=−5 After solving for z, you can substitute the value of z into the first and second equations to find the values of x and y.

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