Problem 44 - Algebra and Trigonometry

Question

Write the system of equations as a matrix equation (see Example 6 ). $$ \left\\{\begin{aligned} 6 x-y+z &=12 \\ 2 x+z &=7 \\ y-2 z &=4 \end{aligned}\right. $$

Step-by-Step Solution

Answer

Matrix equation: \(AX = B\) with \(A = \begin{bmatrix} 6 & -1 & 1 \\ 2 & 0 & 1 \\ 0 & 1 & -2 \end{bmatrix}, X = \begin{bmatrix} x \\ y \\ z \end{bmatrix}, B = \begin{bmatrix} 12 \\ 7 \\ 4 \end{bmatrix}\).

Step 1: Understand the System of Equations

The system of equations provided is: \(6x - y + z = 12\), \(2x + z = 7\), and \(y - 2z = 4\). We need to convert this to a matrix equation of the form \(AX = B\), where \(A\) is the coefficient matrix, \(X\) is the column matrix of variables, and \(B\) is the column matrix of constants.

Step 2: Identify Coefficients and Constants

Extract the coefficients of each variable from the equations: From the first equation: \(6x - 1y + 1z\) gives coefficients \([6, -1, 1]\).From the second equation: \(2x + 0y + 1z\) gives coefficients \([2, 0, 1]\).From the third equation: \(0x + 1y - 2z\) gives coefficients \([0, 1, -2]\). The constants from each equation are 12, 7, and 4, respectively.

Step 3: Formulate the Coefficient Matrix

Construct the coefficient matrix \(A\) using the coefficients from each equation. This results in a matrix:\[A = \begin{bmatrix}6 & -1 & 1 \2 & 0 & 1 \0 & 1 & -2 \\end{bmatrix}\]

Step 4: Formulate the Variable Matrix

The variable matrix \(X\) consists of the variables \(x, y, z\):\[X = \begin{bmatrix} x \ y \ z \end{bmatrix}\]

Step 5: Formulate the Constant Matrix

The constant matrix \(B\) is composed of the constants from the equations:\[B = \begin{bmatrix} 12 \ 7 \ 4 \end{bmatrix}\]

Step 6: Write the Matrix Equation

Combine the matrices \(A\), \(X\), and \(B\) to form the matrix equation \(AX = B\):\[\begin{bmatrix} 6 & -1 & 1 \ 2 & 0 & 1 \ 0 & 1 & -2 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 12 \ 7 \ 4 \end{bmatrix}\]

Key Concepts

Coefficient Matrix

When we talk about a coefficient matrix, we refer to a specific arrangement of numbers that come from the equations in a system. In the given problem, we have three equations, each with variables like x, y, and z.

The coefficients are the numbers that are directly multiplying the variables in the equations. For instance, in the equation \(6x - y + z = 12\), the coefficients are \([6, -1, 1]\).

• The first column in the matrix represents the coefficient of \(x\) from each equation.
• The second column represents the coefficient of \(y\).
• The third column represents the coefficient of \(z\).

Setting up a 3x3 matrix, we place these coefficients in a specific order, reflecting how they align in the original equations. With these, you form the matrix \([A]\), which streamlines solving the equations using matrix methods.

System of Equations

A system of equations consists of multiple equations working together, each with similar variables. Each equation in the system considers the relationships between these variables and constants.

This concept is crucial because it provides a structured way to find where all these considerations meet, typically a solution those variables satisfy simultaneously.

• An equation like \(6x - y + z = 12\) is just one piece of the puzzle.
• Combined with others, they create a holistic picture of how 'x', 'y', and 'z' interact.

By viewing equations as parts of a whole, you'll discover a unified solution through methods like substitution, elimination, or more sophisticated means like matrix algebra.

Matrix Algebra

Matrix algebra provides a framework where we can handle systems of equations using matrices. It shifts from standard algebraic manipulation to an efficient system using arrays of numbers.

Here’s how it benefits:

• Allows for a more concise representation of the problem.
• Simplifies large, complex systems into manageable equations.
• Uses operations on matrices like addition, subtraction, and multiplication.

In solving these systems by matrix algebra, it's often quicker and less prone to error than normal algebra, especially when dealing with numerous variables. With the matrix equation \(AX = B\), matrix algebra offers tools to find the values of \(X\) comprehensively.

Variables and Constants

Variables and constants are foundational elements of equations. Variables like \(x, y,\) and \(z\) are symbols representing unknown values we aim to discover.
Constants, on the other side, are specific numbers providing fixed values within an equation.

• Variables change depending on the relationships defined by the equations.
• Constants remain the same across the calculations as in \(12, 7,\) and \(4\) from our system.

Understanding these elements helps uncover what each equation tells us about the relationships in the system. By isolating and solving for these variables, you will find a unique or set of solutions that satisfy all equations effectively.