Problem 31 - Algebra and Trigonometry

Question

Write each linear system as a matrix equation in the form \(A X=B\), where \(A\) is the coefficient matrix and \(B\) is the constant matrix. $$ \left\\{\begin{array}{l} {x+3 y+4 z=-3} \\ {x+2 y+3 z=-2} \\ {x+4 y+3 z=-6} \end{array}\right. $$

Step-by-Step Solution

Answer

The matrix equation equivalent to the given system of linear equations is \[A X = B, \]where \[A = \begin{bmatrix}1 & 3 & 4 \1 & 2 & 3 \1 & 4 & 3 \end{bmatrix} , X = \begin{bmatrix}x \y \z \end{bmatrix} , B = \begin{bmatrix}-3 \-2 \-6 \end{bmatrix}\]

Step 1: Identify the Coefficients and Constants

In the given system of linear equations\\n\( x + 3y + 4z = -3 \),\n\( x + 2y + 3z = -2 \),\n\( x + 4y + 3z = -6 \),\nWe see that the coefficients of the variables \(x\), \(y\) and \(z\) are:The coefficients for \(x\) are 1, 1, 1. The coefficients for \(y\) are 3, 2, 4. The coefficients for \(z\) are 4, 3, 3. This forms the coefficient matrix \(A\)

Step 2: Write Down the Coefficient Matrix A

The matrix \(A\) is created by arranging the coefficients of the system of linear equation in the order as they appear, with each row corresponding to a separate equation. Thus, the coefficient matrix \(A\) is as follows: \[A = \begin{bmatrix}1 & 3 & 4 \1 & 2 & 3 \1 & 4 & 3 \end{bmatrix}\]

Step 3: Identify the Constant Matrix B

The constants in the right-hand side of the equations are -3, -2 and -6. These form the constant matrix \(B\).

Step 4: Write Down the Constant Matrix B

We create the constant matrix \(B\) by arranging the constants vertically in the matrix as they appear in the system of equations. As such, we have: \[B = \begin{bmatrix}-3 \-2 \-6 \end{bmatrix}\]

Key Concepts

Understanding a Linear System

A linear system consists of one or more linear equations. Each equation represents a straight line. When dealing with multiple variables, it shows their relationship in a space of higher dimensions, such as a plane or a 3D space.
Linear systems can have:

• No solution (the lines are parallel and never intersect),
• Exactly one solution (all lines intersect at a single point),
• Infinitely many solutions (all lines are the same line, overlapping completely).

For the given exercise, we have three equations with three variables: \(x\), \(y\), and \(z\). Together, they form a system of linear equations where each equation must be true simultaneously.

Coefficient Matrix Explained

The coefficient matrix is crucial because it organizes the coefficients of your variables in a tidy manner. This helps in solving the system efficiently using matrix algebra.
In the linear system we are examining:

• The coefficients of \(x\) are each listed first in their respective equations (1, 1, 1).
• The coefficients of \(y\) come next (3, 2, 4).
• The coefficients of \(z\) are listed last (4, 3, 3).

We gather these into a matrix format called the coefficient matrix, \(A\): \[A = \begin{bmatrix} 1 & 3 & 4 \ 1 & 2 & 3 \ 1 & 4 & 3 \end{bmatrix}\]
Each row in this matrix represents a complete equation from the original system, helping maintain the structure and sequence of the equations.

Constant Matrix: What It Is and Why It Matters

The constant matrix, often referred to as \(B\), is an essential component in forming a matrix equation. It consists solely of the constants from each equation, typically found on the right side of the equation.
In our specific example, the constants are \(-3\), \(-2\), and \(-6\). These values are gathered into a single-column matrix:\[B = \begin{bmatrix} -3 \ -2 \ -6 \end{bmatrix}\]
This matrix makes it easier to align with the corresponding equations from the coefficient matrix. Together they allow for straightforward computations using techniques like matrix operations or row reduction to find variable solutions.

Interpreting the System of Equations

A system of equations is essentially a set of equations with multiple variables. Solving this set means finding a common solution that satisfies all of the equations simultaneously.

• The system provided: \(x + 3y + 4z = -3\), \(x + 2y + 3z = -2\), and \(x + 4y + 3z = -6\) is a classic example.
• To solve, we represent the system as a matrix equation using \(AX = B\), where \(A\) is the coefficient matrix and \(B\) is the constant matrix.

The process of solving these equations can reveal whether the system has a single unique solution, no solution, or infinitely many solutions. Utilizing the matrix form particularly allows for solving larger systems efficiently, expanding these methods from manual calculations to computer-assisted solutions.