Problem 58 - College Algebra

Question

Solve using a graphing utility's \(|\operatorname{REF}|\) or \(|\) RREF \(|\) command: $$ \left\\{\begin{array}{c} 2 x_{1}-2 x_{2}+3 x_{3}-x_{4}=12 \\ x_{1}+2 x_{2}-x_{3}+2 x_{4}-x_{5}=-7 \\ x_{1}+\quad \quad x_{3}+x_{4}-5 x_{5}=1 \\ -x_{1}+x_{2}-x_{3}-2 x_{4}-3 x_{5}=0 \\ x_{1}-x_{2}- x_{4}+x_{5}=4 \end{array}\right. $$

Step-by-Step Solution

Answer

The solution will depend on the specific calculator or software used to compute the Row Echelon Form or Reduced Row Echelon Form, but will be a set of values for the variables \(x_1\), \(x_2\), \(x_3\), \(x_4\), and \(x_5\).

Step 1: Set Up the Augmented Matrix

First, set up the augmented matrix. This matrix consists of the coefficients of the unknowns and the constant terms on the right side of each equation. For example, the first row corresponds to the first equation, so it is \([2,-2,3,-1,0,12]\)

Step 2: Enter the Matrix in the Calculator

Next, enter this matrix into the calculator or software. Most calculators or software have a matrix input feature.

Step 3: Compute the Row Echelon Form or Reduced Row Echelon Form

Compute the Row Echelon Form (REF) or Reduced Row Echelon Form (RREF) depending on the provided tool. This transforms the matrix into a form where the solution can be easily read off. Instructions for how to do this will vary depending on the specific calculator or software being used.

Step 4: Interpret the Results

The last column of the matrix now obtained from REF or RREF gives the values of the unknowns in the system of equations. For instance, if you obtained a last row as \([0,0,0,0,1,-3]\), this would represent the equation \(1* x_5 = -3\), so \(x_5 = -3\)

Key Concepts

Augmented Matrix

When you're faced with a system of linear equations, organizing them into a matrix can simplify your work. An augmented matrix combines the coefficients of the variables and the constants from each equation into a single matrix. This is achieved by writing down the coefficients of each variable in a row format for each equation. For example, take an equation,

• \(2x_{1} - 2x_{2} + 3x_{3} - x_{4} = 12\).

The corresponding row in the augmented matrix will be \([2, -2, 3, -1, 0, 12]\). The idea is to line up all variables and include the constants as the last column. With all equations in matrix form, you can apply specific tools to solve them efficiently. The augmented matrix serves as the first step in methods such as Gaussian elimination to simplify and solve systems of linear equations quickly.

Row Echelon Form

Row Echelon Form (REF) is a powerful tool used to simplify the solution of systems of linear equations. REF transforms the augmented matrix into a simpler form by using row operations. In this form:

• Every non-zero row is above any row of all zeros.
• The leading coefficient in each non-zero row, known as a pivot, is always to the right of the leading coefficient in the row above it.
• Although not required, it is typical for leading coefficients to be 1 and for any numbers below a leading 1 to be zero.

Why is this useful? It systematically reduces complex systems to simpler forms that are easier to solve manually or with software tools. Transitioning to REF is often a stepping stone towards the Reduced Row Echelon Form, especially in using calculators or graphing utilities.

Reduced Row Echelon Form

Reduced Row Echelon Form (RREF) builds on the concepts of REF but goes a step further in simplifying matrices. Here, matrices are transformed in such a way to make the solution practically visible from the matrix itself. The conditions for RREF include:

• Each leading coefficient is 1.
• Each leading 1 is the only nonzero element in its column.
• Just like in REF, every leading 1 is to the right of the leading 1 in the row above.

RREF is particularly useful because it simplifies even further than REF, to the point where the solutions to the equations can be read directly from the matrix. For example, an RREF matrix may look like the identity matrix on the left, with solution values appearing on the right-rmost column. Finding RREF of a matrix often requires less manual work when using graphing tools that have built-in RREF functions.

Graphing Utility for Solving Equations

Modern technology, including graphing utilities and software, plays a significant role in solving systems of equations. They can rapidly compute Row Echelon Forms and Reduced Row Echelon Forms, saving time and minimizing manual calculation errors. To use a graphing utility, you typically input the augmented matrix format of your system of equations, as described earlier.

• Software like Matlab, TI-84 calculators, or online tools allow you to input matrices directly.
• These tools include functions like `rref()` or `ref()` to compute the desired echelon forms.
• They then output a matrix that you can interpret to find your variable solutions.

With these utilities, mathematics becomes more manageable, particularly for complex systems that would be difficult to reduce manually. They are key resources for students and professionals needing accurate and efficient solutions.