Problem 12 - Algebra and Trigonometry

Question

Write the system of linear equations represented by the augmented matrix. Use \(x, y,\) and \(z,\) or, if necessary, \(w, x, y\) and \(z,\) for the variables. $$ \left[\begin{array}{rrrr|r} {4} & {1} & {5} & {1} & {6} \\ {1} & {-1} & {0} & {-1} & {8} \\ {3} & {0} & {0} & {7} & {4} \\ {0} & {0} & {11} & {5} & {3} \end{array}\right] $$

Step-by-Step Solution

Answer

The system of linear equations derived from the augmented matrix is: \(4w + x + 5y + z = 6,\)\(w - x - z = 8,\)\(3w + 7z = 4,\)\(11y + 5z = 3.\)

Step 1: Identify the coefficients

In the given augmented matrix, each row represents a linear equation and each column corresponds to the coefficients of a variable in the linear equations. From left to right, the variables are \(w, x, y,\) and \(z.\) The values in the fifth column (right of the vertical line) are the constant terms in the equations.

Step 2: Write the first equation

The first row of the matrix is [4, 1, 5, 1 | 6]. This translates to the equation \(4w + x + 5y + z = 6.\)

Step 3: Write the second equation

The second row of the matrix is [1, -1, 0, -1 | 8]. This translates to the equation \(w - x - z = 8.\)

Step 4: Write the third equation

The third row of the matrix is [3, 0, 0, 7 | 4]. This translates to the equation \(3w + 7z = 4.\)

Step 5: Write the fourth equation

The fourth row of the matrix is [0, 0, 11, 5 | 3]. This translates to the equation \(11y + 5z = 3.\)

Key Concepts

Augmented Matrix

An augmented matrix is a powerful tool used in linear algebra to represent a system of linear equations. It combines the coefficients of the linear equations and the constants from the equations' right-hand sides into a single matrix form. This is achieved by placing the coefficients in the columns and aligning the constants in an additional column on the right, separated by a vertical line.
For example, if you have an augmented matrix like:

• First row: \[4, 1, 5, 1 | 6\]
• Second row: \[1, -1, 0, -1 | 8\]
• Third row: \[3, 0, 0, 7 | 4\]
• Fourth row: \[0, 0, 11, 5 | 3\]

Each row can be translated back into a linear equation using the variables. Here, the columns represent coefficients for variables \(w, x, y,\) and \(z\), showing the relationship between them and the resulting constants.

System of Equations

A system of equations consists of multiple equations that are solved together because they share variables. When equations are intersecting or set together, it creates a system that must be approached together to find a common solution for each variable. In our example from the augmented matrix, we have the following system of equations:

• \(4w + x + 5y + z = 6\)
• \(w - x - z = 8\)
• \(3w + 7z = 4\)
• \(11y + 5z = 3\)

Such a system can have unique solutions, no solutions, or infinitely many solutions. The goal is to identify the set of values for the variables that satisfy all equations simultaneously. Depending on the complexity and coefficients, different methods like substitution, elimination, or matrix operations can be used to solve the system.

Linear Algebra

Linear algebra is the branch of mathematics that deals with vectors, vector spaces (also called linear spaces), linear transformations, and systems of linear equations. It provides the necessary tools to manipulate these mathematical structures effectively.
When solving a system of equations, linear algebra allows us to use matrices to understand the relationships between different variables and their coefficients. The methods used in linear algebra, such as row reduction (Gaussian elimination), help transform matrices into simpler forms to ease finding solutions. In many cases, converting a system into its augmented matrix form facilitates these operations, making complex systems more manageable.

Variables in Equations

Variables in equations are symbols—often letters like \(w, x, y,\) and \(z\)—that represent unknown values we aim to determine. In any system of linear equations, these variables interact through various operations like addition and multiplication, linked by the coefficients present in each equation.
In our initial augmented matrix, each column corresponds to a variable's coefficients. The equation's structure exposes these relationships by tying specific coefficients to their corresponding variable, culminating in a constant value. Solving an equation system involves determining the values of these variables that make all the equations true simultaneously. Understanding how to interpret and manipulate these variables is key when dealing with linear algebra and systems of equations.