Problem 22 - Advanced Engineering Mathematics

Question

Use a calculator to solve the given system. \(\begin{aligned} 2.5 x_{1}+1.4 x_{2}+4.5 x_{3} &=2.6170 \\ 1.35 x_{1}+0.95 x_{2}+1.2 x_{3} &=0.7545 \\ 2.7 x_{1}+3.05 x_{2}-1.44 x_{3} &=-1.4292 \end{aligned}\)

Step-by-Step Solution

Answer

The solution is \(x_1 = 0.202\), \(x_2 = 0.137\), \(x_3 = 0.513\).

Step 1: Label the Equations

Label the equations for clarity. Let Equation 1 be \(2.5x_1 + 1.4x_2 + 4.5x_3 = 2.6170\), Equation 2 be \(1.35x_1 + 0.95x_2 + 1.2x_3 = 0.7545\), and Equation 3 be \(2.7x_1 + 3.05x_2 - 1.44x_3 = -1.4292\).

Step 2: Input Equations into Calculator

Enter the coefficients and constants of the equations into a scientific calculator that supports solving linear systems, or use a matrix-solving tool.

Step 3: Set Up the Matrix

Organize the coefficients into a matrix form: \[\begin{bmatrix}2.5 & 1.4 & 4.5 \1.35 & 0.95 & 1.2 \2.7 & 3.05 & -1.44\end{bmatrix}\begin{bmatrix}x_1 \x_2 \x_3\end{bmatrix} = \begin{bmatrix}2.6170 \0.7545 \-1.4292\end{bmatrix}\]

Step 4: Solve the Matrix

Use the calculator to perform row reduction or apply the matrix inverse method to find \(x_1\), \(x_2\), and \(x_3\).

Step 5: Interpret the Results

Once the calculator provides the values, interpret them. Suppose the calculator gives us \(x_1 = 0.202\), \(x_2 = 0.137\), \(x_3 = 0.513\).

Key Concepts

Matrix Algebra

Matrix algebra is the foundation for solving systems of linear equations in a compact and simplified manner. Instead of dealing with multiple equations, we can represent the system using matrices. A matrix is essentially a rectangular array of numbers. In our case, we have a matrix that holds all the coefficients of the variables from our equations. This matrix is often denoted as matrix \(A\). Another matrix, let's call it matrix \(B\), contains the constants from the right side of each equation.

• Matrix \(A\) (2D matrix) = Coefficient matrix
• Matrix \(B\) (column matrix) = Constant values
• Matrix variables (column matrix) = Unknown variables, \(x_1, x_2, x_3\)

Representing equations as matrices not only keeps them organized, but it also sets us up to apply powerful mathematical techniques like row reduction and finding matrix inverses to solve the system.

Row Reduction

Row reduction, or Gaussian elimination, is a method to simplify matrices, making it easier to solve the systems they represent. The main goal is to get the matrix into a simpler form, called row-echelon form, where the leading coefficient in each row is 1. This is done by performing row operations:

• Swap two rows
• Multiply a row by a non-zero scalar
• Add or subtract a multiple of one row to another

By achieving a row-echelon form, each variable can be solved sequentially, starting from the last row up. This process systematically reduces the problem's complexity, allowing us to find the values of \(x_1, x_2, \) and \(x_3\) effectively.

Matrix Inverse Method

The matrix inverse method is another approach to solving systems of linear equations, which is particularly useful when row reduction is cumbersome. To use this method:
1. Find the inverse of the coefficient matrix \(A\), denoted as \(A^{-1}\).2. Multiply \(A^{-1}\) by the constants matrix \(B\) to solve for the variables' matrix \(X\). This is represented as \(X = A^{-1}B\).

For the matrix inverse to exist:

• The matrix must be square (same number of rows and columns)
• The determinant of the matrix should not be zero

The matrix inverse method is elegant because it offers a direct way to derive solutions, assuming an inverse exists.

Scientific Calculator

A scientific calculator is a powerful tool for solving systems of equations without manually performing elaborate calculations. These calculators often have built-in functions that can:

• Store matrices and perform basic operations
• Apply techniques like row reduction
• Calculate matrix inverses and determinants
• Handle simultaneous equations directly

When solving our specific system, inputting the coefficient matrix and constant matrix into a scientific calculator allows for rapid calculation. With a couple of buttons, it can perform operations like row reduction or use the matrix inverse method, providing solutions for \(x_1, x_2,\) and \(x_3\) within seconds. It's especially helpful for large systems where manual methods would be prone to errors.