Solve each system of equations using marrices (row...

Solve each system of equations using marrices (row operations). If the system has no solution, say that it is inconsistent.
$$
\left\{\begin{array}{r}
{x+2 y+z=1} \\
{2 x-y+2 z=2} \\
{3 x+y+3 z=3}
\end{array}\right.
$$

Key Concepts

Systems of Linear Equations

A system of linear equations consists of multiple linear equations that involve the same set of variables. The goal is to find values for these variables that satisfy every equation simultaneously. This concept is fundamental in algebra and underlies many applications in mathematics and engineering.

Matrix Representation

Matrix representation involves expressing a system of linear equations as a grid of numbers, where each row corresponds to an equation and each column corresponds to the coefficients of a variable. This form simplifies the manipulation and analysis of the system, making it easier to apply systematic solution methods.

Augmented Matrix

An augmented matrix is created by appending the column of constants from the right-hand side of the equations to the coefficient matrix. This compact representation contains all the necessary information of the system and is the starting point for many algorithmic approaches like Gaussian elimination.

Elementary Row Operations

Elementary row operations consist of three types: swapping two rows, multiplying a row by a nonzero scalar, and adding a scalar multiple of one row to another. These operations are crucial because they allow one to transform an augmented matrix into a simpler form without changing the solution set of the system.

Row Echelon Form and Reduced Row Echelon Form

Row echelon form is a simplified structure of a matrix where all nonzero rows are above any rows of all zeros, and the leading coefficient of a row is always to the right of the leading coefficient of the row above it. Reduced row echelon form goes a step further by ensuring that the leading entries are 1 and are the only nonzero entries in their columns, making the system’s solutions more directly visible.

Consistency of a System

The consistency of a system of equations refers to whether or not the system has any solutions. A system is consistent if there exists at least one set of values for the variables that satisfy all equations. Conversely, it is inconsistent if the system leads to a contradiction, such as an equation that simplifies to a statement like 0 = 1.

Transcript

Please give Ace some feedback