Problem 45 - Precalculus: Mathematical for Calculus

Question

Write the system of equations as a matrix equation (see Example 6). $$\left\\{\begin{aligned} 3 x_{1}+2 x_{2}-x_{3}+x_{4} &=0 \\ x_{1} &-x_{3} \quad=5 \\ 3 x_{2}+x_{3}-x_{4} &=4 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system is represented as \( A\mathbf{x} = \mathbf{b} \).

Step 1: Organize the Equations

First, we need to organize the given system of equations: \( 3x_1 + 2x_2 - x_3 + x_4 = 0 \), \( x_1 - x_3 = 5 \), and \( 3x_2 + x_3 - x_4 = 4 \). Notice that some variables are missing in some equations. We will ensure each equation includes all variables by using zero as their coefficients when necessary.

Step 2: Write in Standard Form

Each equation should be in the form of \( ax_1 + bx_2 + cx_3 + dx_4 = e \), where \( e \) is the constant term. Our system becomes: \( 3x_1 + 2x_2 - x_3 + x_4 = 0 \), \( x_1 + 0x_2 - x_3 + 0x_4 = 5 \), and \( 0x_1 + 3x_2 + x_3 - x_4 = 4 \).

Step 3: Identify the Coefficient Matrix

The coefficient matrix is formed by the coefficients of the variables from each equation. Thus, the matrix is: \[ A = \begin{bmatrix} 3 & 2 & -1 & 1 \ 1 & 0 & -1 & 0 \ 0 & 3 & 1 & -1 \end{bmatrix} \]

Step 4: Identify the Variable Vector

The variable vector consists of the variables of the system. It is written as: \[ \mathbf{x} = \begin{bmatrix} x_1 \ x_2 \ x_3 \ x_4 \end{bmatrix} \]

Step 5: Identify the Constant Vector

The constant vector consists of the constant terms from each equation. Write it as: \[ \mathbf{b} = \begin{bmatrix} 0 \ 5 \ 4 \end{bmatrix} \]

Step 6: Form the Matrix Equation

Combine the coefficient matrix, the variable vector, and the constant vector to form the matrix equation. We express the system as: \[ A\mathbf{x} = \mathbf{b} \], which translates to: \[ \begin{bmatrix} 3 & 2 & -1 & 1 \ 1 & 0 & -1 & 0 \ 0 & 3 & 1 & -1 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \ x_3 \ x_4 \end{bmatrix} = \begin{bmatrix} 0 \ 5 \ 4 \end{bmatrix} \]

Key Concepts

System of Equations

A system of equations is a set of two or more equations that share the same set of variables. These equations are solved simultaneously to find the values of the variables that satisfy all the equations at the same time. Let's think of it like a puzzle where the goal is to find the unique combination of variable values that fits all specified conditions. In our example, we have a system of three equations involving four variables:

• \(3x_1 + 2x_2 - x_3 + x_4 = 0\)
• \(x_1 - x_3 = 5\)
• \(3x_2 + x_3 - x_4 = 4\)

Each equation in the system provides a piece of information about the possible values of the variables. Solving this system means finding values for \(x_1, x_2, x_3,\) and \(x_4\) that make all three equations true simultaneously.

Coefficient Matrix

The coefficient matrix is a tool used to represent the system of equations in a more compact form. It consists of the coefficients of variables from each equation aligned in a rectangular array. This matrix is useful because it helps streamline the processes such as substitution or elimination when solving systems of equations.
In our example, the coefficient matrix for the system of equations is:\[A = \begin{bmatrix} 3 & 2 & -1 & 1 \ 1 & 0 & -1 & 0 \ 0 & 3 & 1 & -1 \end{bmatrix}\]

• The rows of the matrix correspond to each equation.
• The columns correspond to each variable \(x_1, x_2, x_3,\) and \(x_4\).

This matrix includes zeroes to account for the absence of certain variables in an equation. It's an essential component of the matrix equation \(A\mathbf{x} = \mathbf{b}\), where it interacts with the variable vector to produce the system's equations.

Variable Vector

The variable vector is a neat way to include all the variables of the system into a single column. It simplifies computations and helps clearly match them to the coefficient matrix and constant vector. The variable vector puts the spotlight on the variables whose values we aim to solve for.
For the current system of equations, the variable vector is:\[\mathbf{x} = \begin{bmatrix} x_1 \ x_2 \ x_3 \ x_4 \end{bmatrix}\]Seeing the variables in this organized structure makes it easier to comprehend how they interact with the coefficients in the matrix multiplication. This way of grouping variables is not only visually appealing but also functional, especially when performing calculations like applying matrix operations.

Constant Vector

The constant vector is composed of the constants on the right side of each equation in the system. It captures all the fixed numeric values in the system, enabling a comprehensive understanding of each equation's target results without losing clarity. For solving or analyzing the equations, the constant vector is crucial because it provides the specific outcomes that the variable and coefficient interactions must achieve.
In our example, the constant vector is:\[\mathbf{b} = \begin{bmatrix} 0 \ 5 \ 4 \end{bmatrix}\]This vector enables us to set up the entire matrix equation \(A\mathbf{x} = \mathbf{b}\), illustrating the goal of each equation. Through this form, we observe how changes in variables impact the outcome, making solution methods like Gaussian elimination or using matrix inverses much more straightforward.