Problem 7 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Write the system of equations corresponding to each augmented matrix. \(\left[\begin{array}{rrr|r}1 & 3 & 2 & 4 \\ 2 & 0 & 0 & 5 \\ 3 & -3 & 2 & 6\end{array}\right]\)

Step-by-Step Solution

Answer

The short answer based on the given step-by-step solution is: The system of linear equations corresponding to the given augmented matrix is: \[ \begin{cases} x + 3y + 2z = 4 \\ 2x = 5 \\ 3x - 3y + 2z = 6 \end{cases} \]

Step 1: Identify Rows and Coefficients

In the given augmented matrix, the rows represent the system of linear equations, and the first three columns represent the coefficients of the variables in each equation. The last column represents the constants on the right side of the equations, which gives us \[ \left[\begin{array}{rrr|r} 1 & 3 & 2 & 4 \\ 2 & 0 & 0 & 5 \\ 3 & -3 & 2 & 6 \end{array}\right] \] Step 2: Write the System of Equations

Step 2: Write the system of equations

Based on the augmented matrix, create one equation for each row with the coefficients and constants identified in the previous step. Our system of linear equations will be: 1) Equation for first row: \( 1x + 3y + 2z = 4 \) 2) Equation for the second row: \( 2x + 0y + 0z = 5 \) 3) Equation for the third row: \( 3x - 3y + 2z = 6 \) Finally, the system of linear equations corresponding to the given augmented matrix is: \[ \begin{cases} x + 3y + 2z = 4 \\ 2x = 5 \\ 3x - 3y + 2z = 6 \end{cases} \]

Key Concepts

Linear algebra

Linear algebra is a branch of mathematics that focuses on studying systems of linear equations and their solutions. One of its primary elements is the examination of vectors and matrices.
Linear algebra simplifies complex mathematical tasks using matrix notation, making calculations more efficient.
When dealing with systems of equations, linear algebra provides a systematic way of organizing and solving problems.

• It allows us to represent linear equations in a compact form.
• It offers tools such as matrix operations to manipulate equations efficiently.
• Concepts like determinants and eigenvalues provide depth in solving these equations.

By applying linear algebra, we're able to tackle real-world problems across science and engineering, where systems with many equations and variables are common.

Augmented matrix

An augmented matrix is a powerful tool used in solving systems of linear equations. It is a compact way to represent multiple equations at once.
In an augmented matrix, each row corresponds to an equation, and columns represent the coefficients of variables. The last column stands for the constants on the right-hand side of each equation.

• The layout helps to visualize and perform operations on multiple equations easily.
• Operations such as row reduction can be directly executed, simplifying the process of finding a solution.
• An augmented matrix integrates additional elements that help in evaluating and solving equations using methods like Gaussian elimination.

This method provides a clear picture of relationships between equations, often leading to faster and more reliable solutions.

Linear equations

Linear equations form the foundation of the system of equations that we work with in algebra. These equations typically involve unknowns that need to be solved.
A linear equation in terms of variables x, y, z can be expressed as:\[ ax + by + cz = d \]where \(a, b, c\) are constants and \(x, y, z\) are the variables.

• One of the main characteristics is that the graph of a linear equation is a straight line.
• In a system, these equations help describe relationships between different variables.
• The solution for a system of linear equations is the point where all lines intersect.

Understanding linear equations is crucial for grasping how to manipulate and solve them using an organized approach, such as using an augmented matrix.