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. $$\left\{\begin{aligned} x_{1}-x_{2}-5 x_{3}+3 x_{4} &=-1 \\ x_{1}+x_{2}+5 x_{3}-3 x_{4} &=0 \\ x_{2}+5 x_{3}-3 x_{4} &=1 \\ x_{1}-2 x_{2}-10 x_{3}+6 x_{4} &=-1 \end{aligned}\right.$$

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.
$$\left\{\begin{aligned} x_{1}-x_{2}-5 x_{3}+3 x_{4} &=-1 \\ x_{1}+x_{2}+5 x_{3}-3 x_{4} &=0 \\ x_{2}+5 x_{3}-3 x_{4} &=1 \\ x_{1}-2 x_{2}-10 x_{3}+6 x_{4} &=-1 \end{aligned}\right.$$

Key Concepts

System of Linear Equations

A system of linear equations comprises two or more equations where each equation is linear in a set of variables. These systems can be solved simultaneously to find the variable values that satisfy all equations simultaneously. Understanding these systems is fundamental in linear algebra as they appear in many fields of science and engineering, and methods to solve them include substitution, elimination, and matrix techniques.

Triangular Form

Triangular form, also known as row echelon form, is a simplified version of a system of linear equations where the equations are arranged so that each successive equation has one more leading zero than the previous one. This structure makes it easier to solve the system by back-substitution, starting from the last equation and working upwards.

Gaussian Elimination

Gaussian elimination is a systematic method used to bring a system of linear equations into triangular or row echelon form. It involves using elementary row operations—swapping, scaling, and adding multiples of rows—to eliminate variables from equations incrementally. This process simplifies the system, making it straightforward to solve for the unknowns.

Consistency of Systems

The consistency of a system of linear equations indicates whether the system has at least one solution or not. A consistent system has one or more solutions, while an inconsistent system has no solutions. Determining consistency is essential before attempting to find a solution, and it can be verified by checking if an equation reduces to an impossible statement during elimination.

Dependent and Independent Systems

When a system is consistent, it can be either dependent or independent. An independent system has exactly one unique solution, meaning the equations provide precisely the necessary constraints. Conversely, a dependent system has infinitely many solutions because the equations are not all independent; some equations are redundant, expressing the same constraint in different forms. This distinction is revealed through the number of pivot positions relative to the number of unknowns during elimination.

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