Problem 60 - Precalculus with Limits

Question

In Exercises 59-62, use the matrix capabilities of a graphing utility to solve (if possible) the system of linear equations. \(\begin{cases} 2x + 3y + 5z = 4 \\ 3x + 5y + 9z = 7 \\ 5x + 9y + 17z = 13 \end{cases}\)

Step-by-Step Solution

Answer

The solution of the system of equations will be \(x=a\), \(y=b\), \(z=c\) where a, b and c are specific values obtained from the matrix multiplication \(x = A^{-1}B\).

Step 1: Matrix Form of System of Equations

Convert the system of linear equations into the matrix equation form, \(Ax = B\), where A is the coefficient matrix, x is the column matrix of variables, and B is the column matrix of constants. The matrix equation will be \(\begin{pmatrix} 2 & 3 & 5 \ 3 & 5 & 9 \ 5 & 9 & 17 \end{pmatrix} \begin{pmatrix} x \ y \ z \end{pmatrix} = \begin{pmatrix} 4 \ 7 \ 13 \end{pmatrix}\)

Step 2: Compute the Inverse of Matrix A

Compute the inverse of the coefficient matrix A using the calculation method by the graphing utility. Let it be denoted by \(A^{-1}\).

Step 3: Solve for Matrix x

To solve for the variable matrix \(x = [x, y, z]^T\), multiply both sides of the matrix equation with the matrix \(A^{-1}\) to get \(x = A^{-1}B\). This will lead us to the solution matrix for the variables x, y, and z.

Step 4: Interpret Matrix x

After obtaining the solution matrix x (let's denote it as [a, b, c]^T), interpret these values back into the variables x, y, and z. Thus, \(x=a, y=b \) and \(z=c\).

Key Concepts

Matrix Equation

A matrix equation is a way to represent a system of linear equations in a compact, structured form. It helps simplify the process of solving complex equations by turning them into matrix notation, which is easy to handle, especially with a graphing utility.

The typical structure of a matrix equation is given by \(Ax = B\), where:

• \(A\) is the coefficient matrix consisting of the coefficients of the variables.
• \(x\) is the variable matrix (or column matrix) containing the variables \(x, y, z,\) etc.
• \(B\) is the constant matrix containing the constants from the right-hand side of each equation.

The matrix equation form offers clarity and simplicity, making it easier to apply mathematical procedures like finding the inverse or performing multiplication to solve the entire system of equations at once.

Inverse Matrix

The inverse matrix is a powerful tool in solving systems of linear equations. For a matrix to be invertible, it must be a square matrix (same number of rows and columns) and have a non-zero determinant.

When you find the inverse of the coefficient matrix \(A\), denoted as \(A^{-1}\), it allows you to solve the equation \(Ax = B\) by isolating \(x\) through multiplication by the inverse matrix. The resulting equation is \(x = A^{-1}B\), providing the solution to the variables in the system.

Using a graphing utility simplifies this process by calculating the inverse matrix quickly and accurately, giving you the solution without the hassle of manual calculation. This is ideal when dealing with larger systems or when ensuring precision is critical.

Coefficient Matrix

In a system of linear equations, the coefficient matrix \(A\) plays a critical role. It contains all the coefficients from the equations in their respective positions. For example, from the system:\[\begin{cases} 2x + 3y + 5z = 4 \ 3x + 5y + 9z = 7 \ 5x + 9y + 17z = 13 \end{cases}\] the coefficient matrix \(A\) would be:\[\begin{pmatrix} 2 & 3 & 5 \3 & 5 & 9 \5 & 9 & 17 \end{pmatrix}\]

• Each row of \(A\) corresponds to the coefficients of one equation.
• Each column represents the coefficients of a single variable across all equations.

Utilizing the coefficient matrix helps in simplifying the manipulation of equations and efficiently solving them using matrix operations. It is an essential structure for using matrix solutions, providing a clean framework to progress towards finding inverse matrices and solutions.

Graphing Utility

A graphing utility, such as a graphing calculator or specialized software, greatly assists in solving systems of linear equations. These tools can automatically perform complex calculations that would otherwise be tedious or error-prone if done by hand.

Here's how a graphing utility aids with matrix operations:

• It quickly converts equations into matrix form.
• Calculates determinants and inverses of matrices effortlessly.
• Provides a visual representation of equations and solutions.

Using a graphing utility streamlines the process of solving systems of equations by handling the heavy-lifting of matrix computations. This allows students to focus on understanding the methods and concepts rather than getting bogged down by numeric details. It's a fantastic learning aid, bridging the gap between theoretical understanding and practical problem solving.