Problem 37 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Write the given system of linear equations in matrix form. \(\begin{aligned} 2 x-3 y+4 z &=6 \\ 2 y-3 z &=7 \\ x-y+2 z &=4 \end{aligned}\)

Step-by-Step Solution

Answer

The given system of linear equations can be written in matrix form as: \(\begin{bmatrix} 2 & -3 & 4 \\ 0 & 2 & -3 \\ 1 & -1 & 2 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 7 \\ 4 \end{bmatrix}\)

Step 1: 1. Identify the Coefficients and Constants

Given linear equations are: \(2x - 3y + 4z = 6\) \(0x + 2y - 3z = 7\) \(1x - 1y + 2z = 4\) Notice that the coefficients of each variable and the constants on the right-hand side are shown.

Step 2: 2. Construct the Coefficient Matrix, Variable Matrix, and Constant Matrix

We will construct three matrices - the coefficient matrix (A) containing the coefficients of the variables, the variable matrix (X) containing the variables, and the constant matrix (B) containing the constants from the right-hand side of the equations. Coefficient matrix (A): \(\begin{bmatrix} 2 & -3 & 4 \\ 0 & 2 & -3 \\ 1 & -1 & 2 \end{bmatrix}\) Variable matrix (X): \(\begin{bmatrix} x \\ y \\ z \end{bmatrix}\) Constant matrix (B): \(\begin{bmatrix} 6 \\ 7 \\ 4 \end{bmatrix}\)

Step 3: 3. Write the System of Equations in Matrix Form

Now that we have A, X, and B matrices, we can represent the given system of linear equations in matrix form as AX = B. \(\begin{bmatrix} 2 & -3 & 4 \\ 0 & 2 & -3 \\ 1 & -1 & 2 \end{bmatrix}\begin{bmatrix} x \\ y \\ z \end{bmatrix} = \begin{bmatrix} 6 \\ 7 \\ 4 \end{bmatrix}\)

Key Concepts

Coefficient Matrix

When we work with a system of linear equations, the coefficient matrix plays a foundational role. Imagine you have a list of ingredients to make different dishes, where each ingredient's quantity corresponds to a specific dish. Similarly, in a system of linear equations, each equation can be thought of as a 'recipe,' and the coefficient matrix consists of 'quantities' representing how much of each variable is used in each equation.

For the set of equations provided in the exercise, the coefficient matrix consists of the numbers in front of the variables. Specifically, from our exercise, we build a matrix that looks like this:
\[\begin{bmatrix}2 & -3 & 4 \0 & 2 & -3 \1 & -1 & 2 \end{bmatrix}\]
Here, the first row corresponds to the coefficients in the first equation, the second row to the second equation, and so on. This matrix organizes all the numerical information related to variables that we need to solve the system.

Variable Matrix

Once the coefficients are neatly arranged, the variables need an organized form as well. This is where the variable matrix comes into play. It's like having a holder for the tools you will need to craft something. In our case, the 'tools' are variables.

In matrix form, the variables of the system are placed into a column, creating the variable matrix. For the exercise we have, the variable matrix is given by:
\[\begin{bmatrix}x \y \z\end{bmatrix}\]
Each variable is assigned to its own row in this matrix. It is important because it establishes the order in which we are evaluating our system and sets the stage for the upcoming calculations to solve for these variables.

Constant Matrix

Moving on to what the equations are set equal to, we have the constant matrix. Think of it as the price tags of the dishes in our earlier analogy - it's what each dish (equation) costs independently.

In the context of our system of linear equations, the constant matrix is made up of the right-hand side numbers from each equation. It looks like this for our given problem:
\[\begin{bmatrix}6 \7 \4\end{bmatrix}\]
This matrix is essential as it represents the 'results' of each 'recipe' when the variables are mixed according to the coefficients. It is our end goal in the system - to find the set of variables that satisfy these constants.

Matrix Representation of Equations

Finally, let's blend all the previous components to form the matrix representation of our system of equations. It's like putting the ingredients, tools, and price tags together to guide the creation of the dishes.

With the matrices for coefficients, variables, and constants (A, X, and B respectively), our system can be elegantly written as an equation: AX = B. For the problem at hand, it translates to:
\[\begin{bmatrix}2 & -3 & 4 \0 & 2 & -3 \1 & -1 & 2 \end{bmatrix}\begin{bmatrix}x \y \z\end{bmatrix} = \begin{bmatrix}6 \7 \4\end{bmatrix}\]
In this compact form, rather than dealing with individual equations, the matrix encapsulates the entire system. This not only makes it easier to handle but also opens up powerful methods for solving the system, such as matrix inversion or Gaussian elimination. And this, in a nutshell, is how a system of linear equations is represented in matrix form - a format that's perfectly primed for efficient solution-finding.