Problem 10 - College Algebra

Question

In Exercises \(9-12,\) write the system of linear equations represented by the augmented matrix. Use \(x, y, z,\) and, if necessary, \(w, x, y,\) and \(z,\) for the variables. \(\left[\begin{array}{rrr|r}7 & 0 & 4 & -13 \\ 0 & 1 & -5 & 11 \\ 2 & 7 & 0 & 6\end{array}\right]\)

Step-by-Step Solution

Answer

The system of linear equations is: \(7x + 4z = -13\), \(y - 5z = 11\), \(2x + 7y = 6\)

Step 1: Identify Variables and Constants from Matrix

The first step of this problem involves identifying each variable corresponding to its coefficient listed in the matrix. In the given matrix, the first, second and third column belong to the variable \(x, y, z\) respectively. The fourth column indicates the constant values for each equation.

Step 2: Write the Linear Equations

Now, write out each equation by associating each coefficient with its corresponding variable. The sign between the variables also matters. The first equation is \(7x + 0y + 4z = -13\), which simplifies to \(7x + 4z = -13\). The second equation is \(0x + y - 5z = 11\), which simplifies to \(y - 5z = 11\). The third equation is \(2x + 7y + 0z = 6\), which simplifies to \(2x + 7y = 6\).

Key Concepts

Augmented Matrix

An augmented matrix is a tool used to effectively represent a system of linear equations. It provides a compact way to handle the equations by organizing the coefficients and constant terms in a matrix format. The layout consists of rows and columns. Each row corresponds to an equation, while each column corresponds to a variable, with an additional column for the constants.
For example, consider the augmented matrix:

• \[\left[\begin{array}{rrr|r}7 & 0 & 4 & -13 \ 0 & 1 & -5 & 11 \ 2 & 7 & 0 & 6\end{array}\right]\]

Here, the first three columns contain the coefficients of the variables in the equations: \(x\), \(y\), and \(z\). The last column is the constant term for each corresponding equation. This left-right separation helps in distinguishing between the variable coefficients and the constants involved.

System of Equations

A system of equations is a collection of two or more equations with the same set of unknowns. Solving a system of equations involves finding the values of the variables that satisfy all the equations simultaneously.
In the context of linear algebra, systems of equations can often be represented by matrices. The given augmented matrix can represent the following system of equations:

• Equation 1: \(7x + 4z = -13\)
• Equation 2: \(y - 5z = 11\)
• Equation 3: \(2x + 7y = 6\)

The goal when solving these is typically to find a single solution, multiple solutions, or to determine that no solution exists. Each equation in the system is a linear equation, which makes the system a linear system. Solving such a system often involves methods like substitution, elimination, or matrix operations.

Variables in Equations

Variables in equations serve as placeholders that represent numbers, and they are crucial for forming and solving equations. In linear equations, these are often denoted by letters such as \(x\), \(y\), \(z\), etc.
When we interpret an augmented matrix, we translate each element in the matrix's rows into coefficients for these variables in their respective equations. Insightfully selecting variables aids in breaking down the system of equations into manageable parts. For the given augmented matrix, we have:

• \(x\) for the first column
• \(y\) for the second column
• \(z\) for the third column

These variables come together to form the overall system of equations. They allow us to express algebraic relations and solve for the values that satisfy all these conditions.

Matrix Representation

Matrix representation is a mathematical technique that assists in expressing linear equations in a structured format. It simplifies the manipulation and solution of systems by organizing them into a matrix. The crucial idea is to align coefficients of variables in the matrix, facilitating operations such as row reduction.
The matrix form can be split into the coefficient matrix and the constants matrix:

• Coefficient Matrix: \(\left[\begin{array}{rrr}7 & 0 & 4 \ 0 & 1 & -5 \ 2 & 7 & 0\end{array}\right]\)
• Constants Matrix: \(\left[\begin{array}{c}-13 \ 11 \ 6\end{array}\right]\)

By representing equations this way, one can apply methods like Gaussian elimination to solve the system efficiently. Overall, matrix representation is a powerful tool in linear algebra, allowing for leveraging computer algorithms to handle large systems effectively.