Problem 76 - Algebra and Trigonometry Real Mathematics, Real People

Question

$$\left\\{\begin{aligned} 2 x+y-z+2 w &=-6 \\ 3 x+4 y &+w=1 \\ x+5 y+2 z+6 w &=-3 \\ 5 x+2 y-z-w &=3 \end{aligned}\right.$$

Step-by-Step Solution

Answer

Without specific computations, the exact solutions for variables \(x, y, z, w\) cannot be determined here. Once arithmetic operations are performed based on the provided steps, the specific solutions can be obtained.

Step 1: Identify The Equations

There are four equations provided: 1) \(2x + y - z + 2w = -6\)2) \(3x + 4y + w = 1\)3) \(x + 5y + 2z + 6w = -3\)4) \(5x + 2y - z - w = 3\)

Step 2: Solve the System Using Matrix Method

The system of equations can also be represented as a matrix. Use Gaussian Elimination or Gauss-Jordan Elimination to solve it.

Step 3: Normalize Row 1

The first step in Gaussian Elimination is to normalize the first row. Divide each element in the first row by the element at \(A[1,1]\) which is '2'.

Step 4: Zero Out Elements Below \(A[1,1]\)

The next step is to make the elements below \(A[1,1]\) zero. This is achieved by multiplying the normalized row 1 by a suitable number and then subtracting it from row 2, row 3, and row 4.

Step 5: Normalize Row 2

Next, Normalize the updated Row 2 using the element at \(A[2,2]\).

Step 6: Zero Out Elements Below and Above \(A[2,2]\)

Make the elements below and above \(A[2,2]\) zero in a similar manner to Step 4.

Step 7: Repeat Steps 5 and 6 for Rows 3 and 4

Repeat the normalizing and zeroing out process for the rest of the rows. After these steps, find yourself with a row-reduced echelon form. The system of equation now is much simpler to solve.

Step 8: Solve The Simplified System

Solve the simplified system of equations, and get the value of each variable. The values of \(x, y, z, w\) are your solutions.

Key Concepts

Gaussian Elimination

Gaussian Elimination is a method used to solve systems of linear equations. It involves using a series of row operations to transform the system into an upper triangular matrix, from which the solutions can easily be determined. This process begins by normalizing the first row to create a leading one, followed by eliminating the coefficients below the leading one. The same steps are then systematically applied to subsequent rows.

Normalization of the First Row

Just like in the exercise, the first step is to make the first element of the first row equal to one, if it isn't already. All other elements in the first row are then adjusted accordingly. This step sets the stage for the upcoming elimination process.

Row Elimination

Next, using the normalized first row, coefficients below the leading one are turned into zeros by performing row operations. This is critical to shaping the upper triangular matrix form, which eventually makes back-substitution straightforward.

• Simplify the coefficients.
• Use row additions or subtractions.
• Progress row by row, in order.

After these operations, the matrix represents a system where each equation builds on the previous one, leading to the final solution when the variables are solved in a reverse order from last to first.

Gauss-Jordan Elimination

The Gauss-Jordan Elimination is an extension of the Gaussian method, taking the process a step further to arrive at reduced row echelon form. In this form, each leading coefficient is one (normalized), and all elements above and below it are zeros.

Zeroing Above and Below

Unlike Gaussian Elimination, which leaves the upper triangle, Gauss-Jordan aims to zero out not just the elements below the leading ones, but also those above. This creates a diagonal of ones across the matrix, simplifying the determination of variable values.

• Further row operations are performed.
• Normalize each row's leading coefficient.
• Make sure all non-leading positions are zero.

By performing these steps, the matrix becomes a series of simple equations (like x = a, y = b, etc.), which can be read off directly without the need for back-substitution, enabling faster and more direct solutions.

Matrix Representation

Matrix representation is a compact and structured way to express systems of linear equations. By arranging coefficients into rows and columns, we obtain matrices, lending themselves well to methodical operations like those used in Gaussian and Gauss-Jordan elimination.

Structure of a Matrix

The first step is to translate the system of equations into a matrix. This involves placing the coefficients of variables in a grid format, with each row representing an equation.

• Equations are mapped into rows.
• Variables and their coefficients become columns.
• The equal signs are represented by vertical lines.

In the exercise, the matrix incorporates constants on the right side, forming an augmented matrix. This layout is pivotal as it keeps the system organized and synchronizes changes across the equations during the elimination process.

Row Reduction

Row reduction is a technique used within methods like Gaussian and Gauss-Jordan elimination. It involves modifying the rows of a matrix to achieve a desired form, which helps in solving the associated system of linear equations.

Performing Row Reduction

The aim is to simplify the matrix such that each row contains fewer nonzero elements than the one above it. To accomplish this, a series of row operations are performed:

• Row additions or subtractions to eliminate coefficients.
• Row multiplications to normalize for a leading one.
• Interchange rows, if necessary, for proper form.

These operations are reflected across the entire row and must maintain the balance of the original equations. Ultimately, row reduction paves the way towards a simplified matrix from which variable values are readily extracted. During these operations, maintaining a systematic and careful approach is crucial in avoiding errors, and in the simplification of the matrix.