Find the complete solution of the linear system, or show that it is inconsistent. { x-2 y+z=

step 1 Here we have given system of equation x minus 2y plus z equals to 3 and then 2x minus 5y plus 6

Find the complete solution of the linear system, or show...

Key Concepts

Consistency of Linear Systems

The consistency of a linear system determines whether the system has at least one solution. After applying Gaussian elimination, if a row is reduced to an impossibility (for example, a row with all zero coefficients but a nonzero constant), the system is inconsistent. Otherwise, the system is consistent and may possess a unique solution or infinitely many solutions.

Free Variables and Parametric Solutions

In cases where a linear system has infinitely many solutions, some variables are not leading variables and can take arbitrary values; these are known as free variables. The complete solution is then expressed in a parametric form, with the free variables acting as parameters that describe the entire solution set.

Linear Systems of Equations

A linear system is a collection of equations where each is linear in the variables. The goal is to find values for these variables that satisfy every equation simultaneously. These systems can have a unique solution, infinitely many solutions, or no solution at all.

Gaussian Elimination

Gaussian elimination is a method used to simplify linear systems by performing a series of elementary row operations. This process transforms the system's augmented matrix into an upper triangular or row echelon form, which makes it easier to solve for the variables or determine the system's consistency.

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