Problem 60 - College Algebra

Question

Solve using a graphing utility's [REP or RREF command: $$\begin{aligned} 2 x_{1}-2 x_{2}+3 x_{3}-x_{4} &=12 \\ x_{1}+2 x_{2}-x_{3}+2 x_{4}-x_{5} &=-7 \\ x_{1}+& x_{3}+x_{4}-5 x_{5}=1 \\ -x_{1}+x_{2}-x_{3}-2 x_{4}-3 x_{5} &=0 \\ x_{1}-x_{2}-& x_{4}+x_{5}=4\end{aligned}$$

Step-by-Step Solution

Answer

The solution to the system of equations can be obtained by interpreting the output matrix returned by the REP or RREF command. However, without a graphing calculator, it is not possible to provide the numerical solutions. Refer to the user manual of the graphing calculator for detailed steps.

Step 1: Create the augmented matrix from the system of equations

An augmented matrix is a matrix obtained by appending the coefficients of the right side of the equations to a coefficient matrix. Here, the coefficient matrix comprises the coefficients of \(x_1\), \(x_2\), \(x_3\), \(x_4\), and \(x_5\). The augmented matrix for this system of equations is: \[\begin{bmatrix} 2 & -2 & 3 & -1 & 0 & 12 \ 1 & 2 & -1 & 2 & -1 & -7 \ 1 & 0 & 1 & 1 & -5 & 1 \ -1 & 1 & -1 & -2 & -3 & 0 \ 1 & -1 & 0 & -1 & 1 & 4 \end{bmatrix}\]

Step 2: Enter the matrix to a graphing calculator

Depending on the model and brand of the calculator, the method of entering the matrix may vary. It is crucial to follow the instruction manual for entering the matrix correctly. Make sure to enter the matrix exactly as given, without making any errors.

Step 3: Use the REP or RREF command

Again, this step would vary based on the type of calculator. Usually, you can find these commands in the 'matrix' menu. Select the matrix that was entered, and then select the REP or RREF command.

Step 4: Interpret the result

The output matrix returned by the REP or RREF command would be the solution to the system of equations. The last column should contain the solutions for \(x_1\), \(x_2\), \(x_3\), \(x_4\), and \(x_5\) in respective order.

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with a set of variables. In this particular problem, we are dealing with five variables: \( x_1, x_2, x_3, x_4, \) and \( x_5 \). The challenge is to find values for these variables that satisfy all the given equations simultaneously.
To achieve this, we can use matrix operations to transform these equations into an augmented matrix. This approach helps streamline the process of finding the solution by using techniques such as row reduction and operations that can be executed efficiently with the help of a graphing calculator or software.

RREF

RREF stands for Row Reduced Echelon Form, a type of matrix related to systems of linear equations. When a matrix is in RREF, it reveals the solutions to the system of equations it represents, in a particularly neat format.
In the RREF, each leading entry in a non-zero row is 1, and all the entries in the column containing a leading entry are 0 except for the leading entry itself. Additionally, each leading 1 is to the right of the leading 1 in the row above it, and rows with all zero elements, if any, are at the bottom. By converting the augmented matrix to its RREF, you can clearly see the solution to your system of equations, often at a glance.

Graphing Utility

A graphing utility, such as a scientific calculator or specialized software, can significantly simplify the work involved in solving a system of equations by converting them into augmented matrices and reducing them to RREF. Once you've entered your matrix correctly into the graphing utility, you can use specific commands (REP or RREF) to perform complex matrix operations with ease.
Using a graphing utility is particularly useful because it minimizes errors that can occur with manual calculations, especially when dealing with large systems or messy fractions. Always ensure you are familiar with your utility's manual for instructions on the correct way to enter matrices and locate the REP or RREF commands.

Solution Interpretation

Once the graphing utility outputs the matrix in RREF, it's time to interpret the results. The last column of the matrix will typically contain the final solutions for each variable of the system of equations.
It's crucial to carefully review this output. For example, if the last column of the RREF matrix of a 5-row system reads \([a, b, c, d, e]\), it implies that \(x_1 = a\), \(x_2 = b\), \(x_3 = c\), \(x_4 = d\), and \(x_5 = e\). Double-checking these results in the original equations can confirm the consistency and correctness of your solution.

• Ensure no rows contradict with a format like \([0, 0, 0, 0, 0 | c]\) where \(c eq 0\), which indicates no solutions.
• Look out for free variables, which give solutions a degree of freedom, implying infinitely many solutions.

Understanding how to extract and verify these solutions is vital for confirming your solution's correctness.