Question

Solve the given set of equations, or else show that there is no solution. $$ \begin{aligned} x_{1}+2 x_{2}-x_{3} &=1 \\ 2 x_{1}+x_{2}+x_{3} &=1 \\\ x_{1}-x_{2}+2 x_{3} &=1 \end{aligned} $$

Short answer

Question: Determine if the given system of linear equations has a unique solution, no solution, or infinitely many solutions. System of linear equations: $$ \begin{cases} x_1 + 2x_2 - x_3 = 1 \\ 2x_1 + x_2 + x_3 = 1 \\ x_1 - x_2 + 2x_3 = 1 \end{cases} $$ Answer: The given system of linear equations has no solution.

Step-by-step solution

Write the augmented matrix

First, we will write the given system as an augmented matrix: $$ \left[\begin{array}{ccc|c} 1 & 2 & -1 & 1 \\ 2 & 1 & 1 & 1 \\ 1 & -1 & 2 & 1 \end{array}\right] $$

Perform row operations to reach row-echelon form

We will perform a series of row operations to simplify the augmented matrix: 1. Swap row 1 and row 2 to get the matrix: $$ \left[\begin{array}{ccc|c} 2 & 1 & 1 & 1 \\ 1 & 2 & -1 & 1 \\ 1 & -1 & 2 & 1 \end{array}\right] $$ 2. Subtract 0.5 times row 1 from row 2. Subtract row 1 from row 3. We get the matrix: $$ \left[\begin{array}{ccc|c} 2 & 1 & 1 & 1 \\ 0 & 1.5 & -1.5 & 0.5 \\ 0 & -2 & 1 & 0 \end{array}\right] $$ 3. Multiply row 2 by 2/3. Add 1/3 times row 2 to row 1. Add row 2 to row 3. We get the matrix: $$ \left[\begin{array}{ccc|c} 2 & 0 & 1 & 1 \\ 0 & 1 & -1 & 1/3 \\ 0 & 0 & 0 & 1/3 \end{array}\right] $$

Check for the number of solutions

The last row of the row-echelon form matrix represents the equation \(0x_1 + 0x_2 + 0x_3 = 1/3\). Since this equation is inconsistent as no values of \(x_1\), \(x_2\), and \(x_3\) can satisfy it, the system has no solution.

Conclusion

As the final row of the row-echelon form matrix represents an inconsistent equation, we can conclude that there is no solution to the given system of linear equations.

Key concepts

Row-Echelon Form

Understanding the row-echelon form is essential when solving a system of linear equations. It's a particular layout for a matrix where all nonzero rows are above any rows of all zeroes, and each leading coefficient (also known as a pivot) of a nonzero row is to the right of the leading coefficient of the row above it. The pivots themselves should be 1, and all elements underneath these pivots must be zero.

To achieve this form, a series of elementary row operations are performed on the augmented matrix of the system. The ultimate goal is to simplify the system to a point where the solutions, if they exist, are easily identifiable. However, if during the process we obtain a row with all zeroes followed by a nonzero number in the augmented part, then we know the system doesn't have a solution, as this contradicts the basic rules of algebra.

Augmented Matrix

An augmented matrix represents a system of linear equations and includes not only the coefficients of the variables but also the constants from the right side of the equations. When written in matrix form, the line separates the coefficient matrix (on the left) from the constants (on the right).

A great benefit of using an augmented matrix is that it streamlines the process of applying row operations. This method neatly combines all important information from the system into a single entity. For the system with equations in the form \(ax + by + cz = d\), the augmented matrix would look like \[\left[\begin{array}{ccc|c}a & b & c & d\end{array}\right]\]. It's this augmented matrix that is manipulated to reach the row-echelon form and solve or determine the nature of the system.

Inconsistent System

A system of equations is labeled inconsistent if there is no possible set of values for the variables that can satisfy all the equations simultaneously. This situation is easily identified in the row-echelon form of an augmented matrix when one or more of the rows effectively states that 0 equals a nonzero constant, such as our example \(0x_1 + 0x_2 + 0x_3 = 1/3\).

A telltale sign of inconsistency is having fewer pivots than there are variables or equations, which indicates there is at least one equation that does not align with the others. When faced with such an inconsistent row, it's a clear indicator that the system cannot be solved and has no solution.

Row Operations

Row operations are tools we use to transform a matrix into simpler forms like the row-echelon form, making it easier to analyze and solve systems of equations. There are three types of basic row operations: swapping two rows, multiplying a row by a nonzero scalar, and adding a multiple of one row to another row.

These operations are core to the Gaussian elimination process, where they're applied sequentially to the augmented matrix with the objective to create a diagonal of 1's (pivots) and zeros below them in the coefficient matrix. Proper use of these operations can reduce a seemingly complex system to one that is straightforward to solve. To maximize understanding and avoid errors, it's essential to apply these row operations carefully and methodically.