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.
$$
\left\{\begin{aligned} x+2 y &=0 \\ x+y &=6 \\ 3 x-2 y &=8 \end{aligned}\right.
$$

Key Concepts

Elementary Row Operations

Elementary row operations—including swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another—are the basic maneuvers used to manipulate an augmented matrix. These operations preserve the solution set of the original system while simplifying the matrix.

Gaussian Elimination

Gaussian elimination is a method used to simplify a matrix to an upper triangular or row-echelon form through a series of row operations. This process systematically reduces the complexity of the system, preparing it for the next step of finding the solution.

Back-Substitution

Back-substitution is the process applied after achieving an upper triangular form through Gaussian elimination. Starting from the last equation, the method involves substituting known variable values back into the previous equations to solve for all unknowns sequentially.

Matrix Representation of a System

Representing a system of linear equations in matrix form allows the equations to be compactly written as a set of coefficients and constant terms. This converts the problem into one involving matrices, making it easier to apply systematic methods such as Gaussian elimination.

System of Linear Equations

A system of linear equations consists of multiple equations involving the same set of variables, where each equation is linear in those variables. Such systems can have a unique solution, no solution, or infinitely many solutions, and solving them is a core task in linear algebra.

Augmented Matrix

An augmented matrix is an extended form of a matrix that includes both the coefficients of the variables and the corresponding constants from the right-hand side of the equations. This representation is particularly useful for applying row operations and solving the system through elimination techniques.

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