The concept of linear equations.

The linear system of equations is a system of equations in which each equation is linear with respect to the unknown quantity (for example, a 2-element equation system of degree one).

Solution:

①Cramer's rule. Solving equations with Cramer's law There are two preconditions: one is that the number of equations is equal to the number of unknowns, and the other is that the determinant of the coefficient matrix is equal to zero. Solving equations with Cramer's law is actually equivalent to solving linear equations by inverse matrix method. It establishes the relationship between the solution of linear equations and its coefficients and constants, but it is necessary to calculate n+1 n-order rows and columns due to the solution. The workload is often very large, so the Cramer's law is often used for theoretical proof and is rarely used for specific solutions.

②Matrix elimination method. The augmented matrix of the linear equations is transformed into a row-simplified step-shaped matrix by the elementary transformation of the row, and the linear equations with the row-simplified step-shaped matrix as the augmented matrix are solved by the original equations. When the equations have solutions, the unknowns corresponding to the unit column vector are taken as non-free unknowns, and the remaining unknowns are taken as free unknowns, and the solution of the linear equations can be found.

For example, to solve a third degree equation and three unknowns:

\(x + 2y + 3z = 9；2x - y + z = 8；3x - z = 3 \)

This will be the matrix entered above:

1 2 3: 9; 2 -1 1: 8; 3 0 -1: 3

Result:

\(x = 2，y = -1，z = 3\)

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