Problem 32 - Algebra for College Students

Question

Subscript notation is frequently used for working with larger systems of equations. For Problems 31-34, use a matrix approach to solve each system. Express the solutions as 4-tuples of the form \(\left(x_{1}, x_{2}, x_{3}, x_{4}\right)\).\(\left(\begin{array}{rl}x_{1}-2 x_{2}+2 x_{3}-x_{4} & =-2 \\ -3 x_{1}+5 x_{2}-x_{3}-3 x_{4} & =2 \\ 2 x_{1}+3 x_{2}+3 x_{3}+5 x_{4} & =-9 \\\ 4 x_{1}-x_{2}-x_{3}-2 x_{4} & =8\end{array}\right)\)

Step-by-Step Solution

Answer

Use row reduction to solve and express the solution as a 4-tuple.

Step 1: Write the System of Equations as a Matrix

First, identify and write down the coefficients and constants of the given system of equations. The system is:\[\begin{align*}&x_1 - 2x_2 + 2x_3 - x_4 = -2 \&-3x_1 + 5x_2 - x_3 - 3x_4 = 2 \&2x_1 + 3x_2 + 3x_3 + 5x_4 = -9 \&4x_1 - x_2 - x_3 - 2x_4 = 8\end{align*}\]This system can be represented in matrix form as \(AX = B\) where:

Step 2: Define Matrix Representation

Identify the matrix \(A\), vector \(X\), and vector \(B\):\[A = \begin{bmatrix}1 & -2 & 2 & -1 \-3 & 5 & -1 & -3 \2 & 3 & 3 & 5 \4 & -1 & -1 & -2\end{bmatrix}, X = \begin{bmatrix} x_1 \ x_2 \ x_3 \ x_4 \end{bmatrix}, B = \begin{bmatrix} -2 \ 2 \ -9 \ 8 \end{bmatrix}\]

Step 3: Row Reduce the Augmented Matrix

Form the augmented matrix \([A|B]\) and use Gaussian elimination to row reduce to row-echelon form:\[\begin{bmatrix}1 & -2 & 2 & -1 & | & -2 \-3 & 5 & -1 & -3 & | & 2 \2 & 3 & 3 & 5 & | & -9 \4 & -1 & -1 & -2 & | & 8\end{bmatrix}\]Perform row operations to simplify the matrix.

Step 4: Solve Using Back Substitution

Once the augmented matrix is in upper triangular form, solve for each variable using back substitution starting from the bottom row. Apply relevant row operations systematically till each variable is expressed in terms of constants.

Step 5: Express the Solution as a 4-Tuple

After solving, express the values of \(x_1, x_2, x_3, \text{ and } x_4\) in the form of a 4-tuple. Suppose a solution is found where \(x_1 = a\), \(x_2 = b\), \(x_3 = c\), \(x_4 = d\), then write the solution as \((a, b, c, d)\).

Key Concepts

Systems of Equations

A system of equations consists of multiple equations that share multiple variables. The goal is to find values for these variables that satisfy all the equations simultaneously.
When dealing with larger systems, such as systems with four equations and four unknowns as seen in the provided exercise, using algebra alone can become cumbersome. Therefore, mathematical methods like matrix equations offer a structured approach.
In this context, each equation in the system is connected, meaning the solution to one can influence the others. Hence, solving systems of equations often involves finding a common solution that works across all given equations.

Gaussian Elimination

Gaussian elimination is a systematic method used to solve systems of linear equations. Through a series of operations, this method transforms a system to simpler forms, making it easier to solve.
The process includes:

• Using row switching to rearrange rows for simpler calculations.
• Scaling rows so the leading coefficients can be 1.
• Eliminating certain variables by adding or subtracting rows.

This reduction leads to an upper triangular matrix, which sets the stage for back substitution, ultimately aiding in solving for the variables one at a time.
Understanding Gaussian elimination is crucial as it forms the backbone of solving many complex systems computationally.

Row Reduction

Row reduction involves performing row operations to simplify a matrix. These operations aim to transform an augmented matrix into row-echelon form or even reduced row-echelon form as needed.
The operations include:

• Swapping two rows.
• Multiplying a row by a nonzero constant.
• Adding or subtracting a multiple of one row to another row.

These actions are taken to clear out the below leading entries and achieve the required forms.
The goal is to streamline the process, minimizing mistakes and making it easy to perform back substitution at the end of the process.

Matrix Representation

Matrix representation refers to rewriting a system of equations as a single matrix equation, typically in the form of \(AX = B\).
Here's what this means:

• \(A\) is the coefficient matrix, containing just the coefficients of the variables.
• \(X\) is the column vector of the variables (e.g., \(x_1, x_2, \ldots\)).
• \(B\) is the constant matrix, comprising the constant terms on the right-hand side of the equations.

This representation simplifies understanding and manipulation of the system, allowing the application of methods such as Gaussian elimination efficiently.
Understanding matrix representation is essential as it is widely used in computer algorithms and solutions for systems of linear equations in advanced mathematics.