Problem 12 - Beginning and Intermediate Algebra

Question

Write a system of linear equations in \(x, y,\) and \(z\) represented by each augmented matrix. $$\left[\begin{array}{rrr|r}1 & 4 & -7 & -11 \\\0 & 1 & 3 & -1 \\\0& 0 & 1 & 6\end{array}\right]$$

Step-by-Step Solution

Answer

The short answer for the given augmented matrix is the system of linear equations: \[\begin{cases} x + 4y - 7z = -11 \\ y + 3z = -1 \\ z = 6 \end{cases}\]

Step 1: Rewrite the augmented matrix as a system of equations

Given the augmented matrix: \(\left[\begin{array}{rrr|r} 1 & 4 & -7 & -11 \\ 0 & 1 & 3 & -1 \\ 0 & 0 & 1 & 6 \end{array}\right]\) We can rewrite this augmented matrix as a system of linear equations: \(1x + 4y - 7z = -11\) \(0x + 1y + 3z = -1\) \(0x + 0y + 1z = 6\)

Step 2: Simplify the system of equations

Now, we can simplify the system of equations by removing any unnecessary terms: \(x + 4y - 7z = -11\) \(y + 3z = -1\) \(z = 6\)

Step 3: Write the system using x, y, and z variables

Finally, we write the system of linear equations in terms of x, y, and z: \[\begin{cases} x + 4y - 7z = -11 \\ y + 3z = -1 \\ z = 6 \end{cases}\]

Key Concepts

Augmented Matrix

An augmented matrix is a convenient representation used in solving systems of linear equations. When you have a set of linear equations, you can organize the coefficients and constants into a matrix form. This matrix has two parts:

• The coefficients of the variables in the equations form the main part of the matrix.
• The constants on the right side of the equations form an additional column, which is the augmentation.

In the example given, the augmented matrix\[\left[\begin{array}{rrr|r}1 & 4 & -7 & -11 \0 & 1 & 3 & -1 \0 & 0 & 1 & 6\end{array}\right]\] corresponds to the linear equations involving the variables \(x\), \(y\), and \(z\). Each row in the augmented matrix corresponds to an equation with the entries showing the coefficients of the variables and the constant term.

Solving Linear Systems

Solving a system of linear equations means finding the values of the variables that satisfy all the equations simultaneously.
Common methods include:

• Substitution: Solving one equation for a variable and substituting it into the other equations.
• Elimination: Adding or subtracting equations to eliminate a variable.
• Augmented Matrix Method: This involves manipulating the augmented matrix to achieve a row-echelon form, ultimately leading to simpler equations to solve.

In the solved exercise, we used the augmented matrix method, which is especially efficient for systems with multiple equations and variables. By simplifying the augmented matrix, you find specific values for each variable as shown in the simplified system of equations:\[\begin{cases}x + 4y - 7z = -11 \y + 3z = -1 \z = 6\end{cases}\]

Variables in Equations

Variables are symbols like \(x\), \(y\), and \(z\) used to represent unknown values in a mathematical expression or equation.
In a system of equations, each equation involves these variables. Depending on the number of equations and their complexity, variables can be independent or dependent on each other.

• Independent variables: Values that can be freely chosen without any restriction from other variables in the system.
• Dependent variables: Values that depend on the independent variables and fulfil the conditions of the system.

In the exercise, after solving and simplifying, we found that \(z = 6\), making \(z\) an independent variable. By substituting \(z\) into the other equations, you can then find \(y\) and \(x\). This process highlights the role of variables in forming and solving linear equations.

Matrix Representation

Matrix representation is a powerful way to handle systems of linear equations. It allows us to visualize, simplify, and solve equations efficiently.
In a matrix representation, such as our example using variables \(x\), \(y\), and \(z\), each element of the matrix directly corresponds to a coefficient in the equation:

• The first column of the matrix corresponds to the coefficients of the variable \(x\).
• The second column corresponds to \(y\), and the third column corresponds to \(z\).
• The vertical line separates these coefficients from the constants, forming the augmented matrix.

This structured organization helps visualize the relationships between equations and variables, making it easier to systematically reduce and solve them, especially useful with techniques like Gaussian elimination or matrix operations.