Problem 48 - Precalculus: Mathematics for Calculus

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

The matrix equation is \(AX = B\) with \(A\), \(X\), and \(B\) as defined steps.

Step 1: Identify Coefficients and Constants

To write the system of equations as a matrix equation, identify the coefficients of the variables \(x\), \(y\), and \(z\), as well as the constants. For the given system:\1. From \(6x - y + z = 12\), the coefficients are \(6\), \(-1\), and \(1\) with constant \(12\).\2. From \(2x + z = 7\), the coefficients are \(2\), \(0\), and \(1\) with constant \(7\).\3. From \(y - 2z = 4\), the coefficients are \(0\), \(1\), and \(-2\) with constant \(4\).

Step 2: Form the Coefficient Matrix \(A\)

Use the coefficients of the variables \(x\), \(y\), and \(z\) from each equation to form the matrix \(A\).The coefficient matrix \(A\) is:\[A = \begin{bmatrix} 6 & -1 & 1 \ 2 & 0 & 1 \ 0 & 1 & -2 \end{bmatrix}\]

Step 3: Form the Variable Column Matrix \(X\)

Express the variables \(x\), \(y\), and \(z\) as a column matrix \(X\).The variable matrix \(X\) is:\[X = \begin{bmatrix} x \ y \ z \end{bmatrix}\]

Step 4: Form the Constant Column Matrix \(B\)

List the constants on the right-hand side of each equation as a column matrix \(B\).The constant matrix \(B\) is:\[B = \begin{bmatrix} 12 \ 7 \ 4 \end{bmatrix}\]

Step 5: Write the Matrix Equation

Now combine the matrices \(A\), \(X\), and \(B\) into a matrix equation of the form \(AX = B\).The matrix equation is:\[\begin{bmatrix} 6 & -1 & 1 \ 2 & 0 & 1 \ 0 & 1 & -2 \end{bmatrix} \begin{bmatrix} x \ y \ z \end{bmatrix} = \begin{bmatrix} 12 \ 7 \ 4 \end{bmatrix}\]

Key Concepts

Understanding Systems of Equations

A system of equations is a set of two or more equations that have common variables and are supposed to be solved simultaneously. Imagine each equation as a slice of reality, and our task is to find a point where all these slices meet.
Solving these systems means you are on a quest to find those unique values of the variables that satisfy all equations involved. Here's why it stands out:

• Interdependencies: Each equation interconnects with others through common variables, creating a mathematical web.
• Simultaneous Solutions: We find values that satisfy all conditions at once, creating a "meeting point" in this web of equations.
• Real-world Applications: From engineering to economics, these systems help solve complex problems with multiple influencing factors.

Solving systems can involve substitution, elimination, or using graphical methods. However, when matrices come into play, they offer us a structured way to handle these systems more efficiently.

Matrix Notation Essentials

Matrix notation allows us to represent systems of equations in a neat and organized manner. The magic lies in translating complex sets of equations into simpler matrix elements.
A matrix is essentially a rectangular array of numbers arranged in rows and columns. Here's how they fit into our equation-solving journey:

• Coefficient Matrix: Known as matrix \(A\), it holds the coefficients of each equation's variables. It acts like an address book for our unknowns.
• Variable Matrix: Denoted by \(X\), it's a column matrix holding the variables \(x\), \(y\), and \(z\).
• Constant Matrix: Shown as matrix \(B\), this contains the constants from each equation, gathering them onto one side.

When we use matrix notation, we are simply substituting these arrays into a format \(AX = B\). This not only provides clarity but also opens up efficient computational methods to solve these systems, especially when dealing with larger sets of equations.

Linear Algebra in Action

Linear algebra is the branch of mathematics that provides us with the tools to work effectively with matrix equations. It unlocks the potential of matrices to solve systems of equations and much more. Here's why linear algebra is such a powerhouse:

• Matrix Operations: This includes addition, subtraction, and multiplication of matrices, allowing us to manipulate and solve equations efficiently.
• Solving Systems with Inversion: By finding the inverse of a matrix, linear algebra offers methods to solve equations quickly.
• Deeper Insights: Concepts like vector spaces, eigenvectors, and eigenvalues offer rich insights into the nature and solutions of systems.

By using linear algebra, we're able to explore solutions from different angles and heights. This broad view is not only useful for simple solving, but also for understanding the inherent structure and behavior of complex equations in various fields.