Question

Use matrices to solve the system of equations (if possible). Use Gaussian elimination with back-substitution or Gauss-Jordan elimination. $$ \left\\{\begin{aligned} 2 x+10 y+2 z &=6 \\ x+5 y+2 z &=6 \\ x+5 y+z &=3 \\ -3 x+15 y-3 z &=-9 \end{aligned}\right. $$

Short answer

The given system of equations has no solution

Step-by-step solution

Create the Augmented Matrix

The system of equations can be written as the matrix: \[ \begin{bmatrix} 2 & 10 & 2 & 6 \ 1 & 5 & 2 & 6 \ 1 & 5 & 1 & 3 \ -3 & 15 & -3 & -9 \ \end{bmatrix} \].

Apply Gaussian Elimination

Now start applying the Gaussian elimination. First, divide the first row by 2 to get 1 as the leading coefficient. Subtract the first row from the second row. And, add the first row to the fourth row. After these operations, the resulting matrix is \[ \begin{bmatrix} 1 & 5 & 1 & 3 \ 0 & 0 & 1 & 3 \ 1 & 5 & 1 & 3 \ 0 & 20 & 0 & 0 \ \end{bmatrix} \].

Continue with Gaussian Elimination

Subtract the first row from the third row. The resulting matrix is: \[ \begin{bmatrix} 1 & 5 & 1 & 3 \ 0 & 0 & 1 & 3 \ 0 & 0 & 0 & 0 \ 0 & 20 & 0 & 0 \ \end{bmatrix} \]. We can see here a contradiction in the last row, which says that 20y=0, so there's no solution to this system of equations.

Conclude Solution

Since the Gaussian elimination resulted in a contradiction, it is concluded that the system of equations has no solution.

Key concepts

Gaussian Elimination

Gaussian elimination is a method used to solve systems of linear equations. It involves transforming the given system into an upper triangular matrix. The process is done through a series of row operations that simplify the system while maintaining the same solutions. These operations include swapping rows, multiplying a row by a non-zero scalar, and adding or subtracting a multiple of one row from another.

To begin with Gaussian elimination, we first write the system of equations in an augmented matrix form. This matrix includes the coefficients of the variables plus an extra column for the constants on the right side of the equations. The goal is to form zeros below each pivot (leading coefficient) as we move down the matrix.

In the example given, the matrix was transformed into an upper triangular form. However, a contradiction appeared when 20y = 0 was reached. This contradiction (20 times nothing equals zero but 20 is non-zero) signifies that the system does not have a solution.

Gauss-Jordan Elimination

Gauss-Jordan elimination is an extension of Gaussian elimination. While Gaussian elimination stops once an upper triangular form is achieved, Gauss-Jordan elimination continues to simplify the matrix further into reduced row-echelon form (RREF). This technique aims to transform the matrix so each leading term in a row is set to one, and all other entries in that column are zeros.

One of the primary advantages of Gauss-Jordan elimination is that it can immediately provide the solution to the system of equations through its resulting matrix form. If it's successful, you end up with a diagonal matrix which very clearly presents the solutions of the variables involved.

In practice, if used in the given problem, Gauss-Jordan would not be able to overcome the fundamental issue of the contradiction found in Gaussian elimination. The contradiction of 20y=0, corresponding to a linearly dependent system, results in no solution, confirming the understanding that both methods reveal consistent results concerning the system’s solvability.

Systems of Equations

Systems of equations involve finding values for variables that satisfy multiple equations at once. They can be linear (all variables to the first power) or non-linear. Solving systems of equations is fundamental across various fields including engineering, physics, economics, and computer science.

For linear systems, solving methods like substitution, elimination, or matrix methods (including Gaussian and Gauss-Jordan) are used extensively. These methods can help find unique solutions, infinite solutions, or identify when no solutions exist.

In the provided example, the system turned out to be inconsistent, indicated by its row operations resulting in logical contradictions. Such inconsistency often means that the original equations represent parallel planes or lines, which never intersect at a common point. Understanding these outcomes aids significantly in practical predictions and problem solving.