Apply elementary roes operations to a matrix to solve the...

Apply elementary roes operations to a matrix to solve the system of equations. If there is no solution, state that the system is inconsistent.
$$\left\{\begin{array}{c}-x+y=2 \\7 x-4 y=-2\end{array}\right.$$

Key Concepts

Elementary Row Operations

These are operations applied to the rows of a matrix that include swapping rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another. They are essential tools for solving systems of linear equations, as they allow the transformation of a matrix into a simpler form without changing the solution set of the system.

Augmented Matrix

An augmented matrix is a compact representation that combines the coefficients of a system of linear equations with the constants from the right-hand side of each equation. This format simplifies the process of applying elementary row operations to solve the system by focusing on the matrix rather than the equations individually.

System of Linear Equations

A system of linear equations consists of multiple equations that share the same set of variables. The goal is to find values for these variables that satisfy all equations simultaneously. These systems can be represented in matrix form, facilitating various solution methods such as elimination and substitution.

Consistency and Inconsistency in Linear Systems

A linear system is said to be consistent if it has at least one solution, meaning there exists a set of values for the variables that satisfies all equations. Conversely, the system is inconsistent if no such set exists. In the process of using row operations, certain outcomes, such as ending up with a contradictory equation (like 0 = nonzero constant), indicate that the system is inconsistent.

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