Question

Find the best approximation to a solution of each of the following systems of equations. a. \(\begin{aligned} x+y-z &=5 \\ 2 x-y+6 z &=1 \\ 3 x+2 y-z &=6 \\\\-x+4 y+z &=0 \end{aligned}\) b. \(3 x+y+z=6\) \(2 x+3 y-z=1\) \(2 x-y+z=0\) \(3 x-3 y+3 z=8\)

Short answer

Both systems may not have an exact solution; analyze approximate solutions using Gaussian elimination.

Step-by-step solution

Analyze System A

For system \( A \), convert the equations into an augmented matrix form:\[\begin{bmatrix}1 & 1 & -1 & | & 5 \2 & -1 & 6 & | & 1 \3 & 2 & -1 & | & 6 \-1 & 4 & 1 & | & 0\end{bmatrix}\]

Perform Gaussian Elimination on System A

Apply Gaussian elimination to reach Row Echelon Form (REF). Start by eliminating the first element below the pivot in the first column using row operations.

Analyze Reduced System from System A

After performing Gaussian elimination, check if the system allows for back substitution and obtain the variables' values if possible.

Analyze System B

Convert the equations of system \( B \) into an augmented matrix:\[\begin{bmatrix}3 & 1 & 1 & | & 6 \2 & 3 & -1 & | & 1 \2 & -1 & 1 & | & 0 \3 & -3 & 3 & | & 8\end{bmatrix}\]

Perform Gaussian Elimination on System B

Apply Gaussian elimination to the augmented matrix of system \( B \) to reach Row Echelon Form (REF) as much as possible.

Analyze Reduced System from System B

From the reduced system, check the consistency of equations and solve the system if possible using back substitution.

Key concepts

Augmented Matrix

When solving a system of linear equations, an augmented matrix is often utilized to simplify the process. It combines the coefficients of the variables and the constants from each equation into a single matrix. By doing this, it transforms the algebraic representation into a more compact and manageable form.
An augmented matrix is written by aligning the coefficients of the variables into columns and the constants into a separate last column, separated by a bar. For example, in system A of the given exercise, the equations are transformed into:

• The first row represents the equation: \( 1x + 1y - 1z = 5 \)
• The additional rows follow similarly, keeping the order of the equations consistent.

Transforming equations into an augmented matrix is often the first step when using the Gaussian elimination method to solve them.

Row Echelon Form

Row Echelon Form (REF) is a key objective when performing Gaussian elimination on an augmented matrix. This form makes it easier to solve the system by creating a stair-step pattern of zeros below each pivot position. A row in REF typically has a leading one, called a pivot, and zeros below it, simplifying later steps like variable isolation.
To achieve REF, we use row operations, including:

• Swapping two rows.
• Multiplying a row by a non-zero scalar.
• Adding or subtracting a scalar multiple of one row to another.

Through these steps, matrices from systems A and B are systematically transformed to uncover any dependencies or inconsistencies. Obtaining REF is essential for establishing the solvability of an equation.

System of Equations

A system of equations is a set of two or more equations that have common variables. The primary goal is to find the values of these variables that satisfy all the equations simultaneously. It can appear daunting, especially when there are multiple variables and equations involved. To simplify the challenge, we convert the system into an augmented matrix.
In the exercise, two different systems are presented:

• System A: Contains four equations with three variables.
• System B: Contains four equations with three variables.

The Gaussian elimination method is a powerful technique to resolve such systems by converting the complex set of equations into simpler forms like the row echelon matrix. This often enables you to discern consistent solutions, revealing if a unique solution, no solution, or infinitely many solutions exist.

Back Substitution

Back substitution is the final step in solving a system of equations after the matrix is reduced to Row Echelon Form (REF). It involves solving for the variables starting from the last row, which typically gives the value of one variable, and substituting that value back into the equations above it.
The process is straightforward but crucial for determining the precise values of each variable:

• Look for the row with one variable and solve it.
• Take the value obtained and substitute it into the preceding row to solve for another variable.
• Continue this process until all variables are found.

After performing Gaussian elimination on systems from the exercise and reaching REF, you'll apply back substitution to derive explicit solutions for the systems, provided they have a viable solution. It's the wrap-up stage where all previous efforts come together to provide the solution set for the original system of equations.