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.
$$\left\{\begin{array}{rr}x+y+z= & 4 \\ 2 x-4 y-z= & -1 \\ x-y= & 2\end{array}\right.$$

Key Concepts

Back Substitution

Back substitution is a method used to solve a system of equations that has been transformed into triangular form. Starting with the equation that contains only one variable, one solves for that variable and then substitutes this result into the preceding equations. This process is repeated until all variables have been solved, leading to the complete solution of the original system.

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The objective is to find the values of these variables that satisfy all the given equations simultaneously. This is a fundamental concept in algebra and linear algebra that has applications in various fields such as engineering, physics, and computer science.

Triangular Form

Triangular form, often achieved via Gaussian elimination, refers to rewriting a system of linear equations so that the system has a staircase pattern with zeros below the main diagonal in its matrix representation. This form simplifies solving the system by allowing back substitution, where one begins solving for the variables one at a time starting from the equation with only one variable.

Gaussian Elimination

Gaussian elimination is a systematic method used to solve systems of linear equations. It involves a sequence of operations—including swapping equations, scaling equations, and adding or subtracting equations—to eliminate variables and reduce the system to triangular form. This process helps in determining whether the system is consistent and, if so, whether it has a unique solution or infinitely many solutions.

Classification of Linear Systems

Linear systems can be classified based on their solutions. A consistent system has at least one solution, while an inconsistent system has none. Within consistent systems, an independent system has a unique solution whereas a dependent system has infinitely many solutions. This classification is important for understanding the behavior of the system and the nature of its solutions.

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