For linear-equation system, variable equals determinant value divided by resultant-determinant value {Cramer's rule} {Cramer rule}.
If two equations contain unknown raised to power, eliminate unknown from both equations by substitution {dialytic method}.
To eliminate a term {elimination, equation}, subtract one equation from another equation. If needed, multiply equation by coefficient or variable before subtracting.
Dividing equations by coefficients and subtracting equations {Gauss-Jordan elimination} can solve equation systems.
process
Divide first row by first-variable coefficient {pivot element}, so first-variable coefficient is one. For other rows, subtract multiple of first row to make first-variable coefficient equal zero, and replace row with resulting row.
Divide new second row by second-variable coefficient, so second-variable coefficient is one. For other rows, subtract multiple of second row to make second-variable coefficient equal zero, and replace row with resulting row.
Follow same steps for all rows. Use pivoting to avoid dividing by zero.
result
All rows begin with variable with coefficient equal one. All rows begin with different variables: row n begins with nth variable.
To solve equation systems, multiply {multiplier method} one equation by a scalar to make unknown's coefficient the same as unknown's coefficient in a second equation. Then subtract first equation from second equation to eliminate term with the unknown. Multiplier method does not change resultant determinant.
Interchanging rows {partial pivoting} or interchanging rows and columns {full pivoting} can put term to eliminate on the diagonal {pivoting in equation solving}. Typically, pivot is largest term.
To solve linear-equation systems, sum all linear-equation powers to derive a power function and then find power-function minimum {power function, linear equations}.
To solve equation systems, rearrange equation terms to have only one variable on equation left side {substitution, equation}. In second equation containing that variable, substitute first-equation right side for variable, to eliminate variable from second equation. This is an example of replacing whole by sum of its parts.
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Date Modified: 2022.0225