In Exercises 49-54 , use matrices to solve the system of...

In Exercises $49-54$ , use matrices to solve the system of equations, if possible. Use Gaussian elimination with back-substitution.
$$
\left\{\begin{array}{l}{-x+y=-22} \\ {3 x+4 y=4} \\ {4 x-8 y=32}\end{array}\right.
$$

Key Concepts

Row Operations

Row operations are the fundamental manipulations used on the rows of an augmented matrix to simplify it. These operations include swapping rows, multiplying a row by a nonzero scalar, and adding or subtracting multiples of rows from one another. They are essential in transforming the matrix into an upper triangular form or reduced row-echelon form without changing the solution set.

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations by systematically applying row operations to an augmented matrix so that it reaches an upper triangular form. This form of the matrix makes it easier to identify the solutions of the system or determine if the system has no or infinitely many solutions.

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. The aim is to find values for these variables that satisfy all equations simultaneously. Such systems can have a unique solution, infinitely many solutions, or no solution, and are foundational in algebra and applied mathematics.

Augmented Matrix

An augmented matrix is a compact form of writing a system of linear equations where the coefficients and constant terms are arranged into a matrix. This representation simplifies the use of systematic methods, such as Gaussian elimination, to solve the system by treating the matrix with row operations.

Back-Substitution

Back-substitution is a technique used as the final step after applying Gaussian elimination. Once the augmented matrix is in an upper triangular form, the variables are solved starting from the last equation and then substituted back into the previous equations to determine all variable values.

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