In Exercises 21–38, solve each system of equations using...

In Exercises 21–38, solve each system of equations using matrices. Use Gaussian elimination with back-substitution or Gauss-Jordan elimination.
$$\left\{\begin{array}{r}{3 a-b-4 c=3} \\ {2 a-b+2 c=-8} \\ {a+2 b-3 c=9}\end{array}\right.$$

Key Concepts

Back-Substitution

Back-substitution is the process of solving for the unknown variables in a system of linear equations after the matrix is reduced to an upper triangular form. Starting from the last equation, each variable is solved and then substituted back into the previous equations to simplify and solve them sequentially.

Gauss-Jordan Elimination

Gauss-Jordan elimination is an extension of Gaussian elimination that further refines the augmented matrix to the reduced row echelon form. In this form, each leading entry is 1 and is the only nonzero entry in its column, allowing the solution of the system to be read directly without the need for back-substitution.

Matrix Representation of Systems

Matrices are used to represent the coefficients and constants of linear systems in a compact form, known as augmented matrices. This representation simplifies the manipulation of systems of equations, making it easier to apply systematic solution techniques like Gaussian elimination.

System of Linear Equations

A system of linear equations is a collection of two or more equations involving the same set of variables. The goal is to find the values of these variables that satisfy all equations simultaneously. This concept forms the basis for many problems in algebra and can be represented succinctly using matrices.

Gaussian Elimination

Gaussian elimination is a method for solving systems of linear equations by applying a series of elementary row operations to convert the augmented matrix into an upper triangular form. Once in this form, back-substitution is used to determine the values of the unknowns progressively from the last equation upward.

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