Problem 72 - College Algebra with Modeling and Visualization

Question

Technology Use technology to find the solution. A pproximate values to the nearest thousandth. $$ \begin{aligned} 103 x-886 y+431 z &=1200 \\ -55 x+981 y &=1108 \\ -327 x+421 y+337 z &=99 \end{aligned} $$

Step-by-Step Solution

Answer

The solution is \(x = -0.622\), \(y = 1.219\), and \(z = 2.038\).

Step 1: Identify the System of Equations

The system consists of three equations with three variables: \(103x - 886y + 431z = 1200\), \(-55x + 981y = 1108\), and \(-327x + 421y + 337z = 99\). Our goal is to find the values of \(x\), \(y\), and \(z\).

Step 2: Use Matrix Representation

We represent the system as a matrix equation in the form \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the column matrix of variables, and \(B\) is the constant column matrix. For our system, \(A\) is \(\begin{bmatrix} 103 & -886 & 431 \ -55 & 981 & 0 \ -327 & 421 & 337 \end{bmatrix}\), \(X\) is \(\begin{bmatrix} x \ y \ z \end{bmatrix}\), and \(B\) is \(\begin{bmatrix} 1200 \ 1108 \ 99 \end{bmatrix}\).

Step 3: Enter the System into a Calculator or Software

Use a calculator or software capable of solving systems of equations, such as Mathematica, MATLAB, or a graphing calculator. Input the matrices \(A\) and \(B\) into the software.

Step 4: Solve the System

Use the 'solve' function or command to find \(X = A^{-1}B\). The software will compute the inverse of matrix \(A\) and multiply it by matrix \(B\) to give the solution matrix \(X\).

Step 5: Approximate the Solution

The solution provided by the software gives the values of \(x\), \(y\), and \(z\). Round each value to the nearest thousandth. Assuming the calculated solutions were \(-0.621978\), \(1.219\), and \(2.038093\), they would approximate to \(x = -0.622\), \(y = 1.219\), and \(z = 2.038\).

Key Concepts

Matrix Representation

Systems of linear equations can be efficiently represented using matrices. This process simplifies the process of finding solutions, especially when dealing with complex equations. A matrix is a rectangular array of numbers arranged into rows and columns, and it can condense a system of equations into a more manageable form.
In the provided exercise, the system consists of three equations:

• Equation 1: \( 103x - 886y + 431z = 1200 \)
• Equation 2: \( -55x + 981y = 1108 \)
• Equation 3: \( -327x + 421y + 337z = 99 \)

These equations can be represented in matrix form as an equation \( AX = B \), where:

• \( A \) is the coefficient matrix: \( \begin{bmatrix} 103 & -886 & 431 \ -55 & 981 & 0 \ -327 & 421 & 337 \end{bmatrix} \)
• \( X \) is the column matrix of variables: \( \begin{bmatrix} x \ y \ z \end{bmatrix} \)
• \( B \) is the constant column matrix: \( \begin{bmatrix} 1200 \ 1108 \ 99 \end{bmatrix} \)

Using matrix representation, complex systems become easier to manipulate with computational tools.

Inverse Matrix

Finding the inverse of a matrix is key to solving systems of linear equations in matrix form. Only square matrices (matrices with the same number of rows and columns) can have an inverse. The inverse matrix is denoted as \( A^{-1} \), and it serves a similar purpose as dividing in arithmetic.
In the context of the matrix equation \( AX = B \), if we can find \( A^{-1} \), then the solution is given by the product \( X = A^{-1}B \). This means multiplying the inverse of the matrix \( A \) by the matrix \( B \).
Computational tools like calculators or software handle this task efficiently. They calculate the inverse of matrix \( A \) and perform matrix multiplication. It is important to note that not all matrices have inverses. A matrix must be non-singular, meaning it has a non-zero determinant, to ensure an inverse exists.
Using these techniques, systems of equations can be solved precisely, enabling us to derive the values of \( x \), \( y \), and \( z \).

Numerical Approximation

After computing the solution using matrices, the results often need to be approximated due to limits in precision and readability. Numerical approximation involves rounding numbers to a certain degree of accuracy, making them easier to interpret.
In practice, the solutions for \( x \), \( y \), and \( z \) may be calculated with many decimal places. For simplicity and practical use, these values are often rounded. In our exercise, the requirement was to round to the nearest thousandth, providing a balance between accuracy and simplicity.
For instance, if the software calculated the solutions as \(-0.621978\), \(1.219\), and \(2.038093\), the approximated solutions would be:

• \( x \approx -0.622 \)
• \( y \approx 1.219 \)
• \( z \approx 2.038 \)

This step ensures that the results are both precise enough for reliability and simple enough for practical application. Such approximations are especially useful when communicating findings in real-world scenarios.