Put each system of linear equations into triangular form and...

Put each system of linear equations into triangular form and solve the system if possible. Classify each system as consistent independent, consistent dependent, or inconsistent. $$\left\{\begin{aligned} x_{1}-x_{3} &=-2 \\ 2 x_{2}-x_{4} &=0 \\ x_{1}-2 x_{2}+x_{3} &=0 \\-x_{3}+x_{4} &=1 \end{aligned}\right.$$

Put each system of linear equations into triangular form and solve the system if possible. Classify each system as consistent independent, consistent dependent, or inconsistent.
$$\left\{\begin{aligned} x_{1}-x_{3} &=-2 \\ 2 x_{2}-x_{4} &=0 \\ x_{1}-2 x_{2}+x_{3} &=0 \\-x_{3}+x_{4} &=1 \end{aligned}\right.$$

Key Concepts

Systems of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. The solution to the system is the set of variable values that satisfies all equations simultaneously. Understanding these systems is fundamental in linear algebra as they model numerous practical problems in science and engineering.

Triangular Form

Triangular form refers to the arrangement of a system (or its augmented matrix) into an upper triangular shape where each successive equation has one less variable than the previous equation. This form significantly simplifies solving the system, as it allows one to solve for the variables using back-substitution.

Gaussian Elimination

Gaussian elimination is a method that involves performing a sequence of elementary row operations (such as row swapping, scaling, and adding multiples of one row to another) to convert the system of linear equations into triangular (or row echelon) form. This process is essential for systematically solving the system and identifying any potential inconsistencies or dependencies among the equations.

System Classification

Systems of linear equations are classified based on the existence and uniqueness of their solutions. A consistent independent system has a unique solution, a consistent dependent system possesses infinitely many solutions because some equations are redundant, and an inconsistent system has no solution due to contradictory equations. This classification is determined by analyzing the positions of pivots in the triangularized or reduced matrix form.

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