Problem 42 - College Algebra and Calculus: An Applied Approach

Question

Write the augmented matrix for the system of linear equations. $$ \left\\{\begin{array}{rr} 2 x+3 y-z= & 8 \\ y+2 z= & -10 \\ x-2 y-3 z= & 21 \end{array}\right. $$

Step-by-Step Solution

Answer

The augmented matrix of the given system of linear equations is: \[ \begin{bmatrix} 2 & 3 & -1 & 8 \ 0 & 1 & 2 & -10 \ 1 & -2 & -3 & 21 \end{bmatrix} \].

Step 1: Identify and Align Coefficients

Rewrite each equation so that like terms are all aligned. The order of terms should be the same across all equations: \[2x + 3y - z = 8 \] \[0x + y + 2z = -10 \] \[x - 2y - 3z = 21 \] Notice the zero coefficient in the second equation for \(x\) because \(x\) isn't mentioned in that equation.

Step 2: Write Out the Matrix Without Variables

Now, we create a matrix where each row corresponds to each equation, and the columns correspond to the coefficients in front of \(x\), \(y\), and \(z\), respectively. The last column is after the = sign. The matrix will look like this: \[\begin{bmatrix}2 & 3 & -1 & 8 \0 & 1 & 2 & -10 \1 & -2 & -3 & 21 \end{bmatrix}\].

Key Concepts

System of Linear Equations

A system of linear equations consists of two or more linear equations involving the same set of variables. Solving these systems means finding the values of the variables that satisfy all equations simultaneously. For example, in our exercise, we have a system of three equations with three variables: \(x\), \(y\), and \(z\). Each equation gives a condition that relates the variables to each other, and we need to find the point (or points) where all these conditions intersect.Understanding how to represent these systems appropriately and solve them is essential for a variety of applications in science, engineering, and economics. The key is to recognize the relationships and constraints each equation places on the variables.

Matrix Representation

Matrix representation is a powerful tool in solving systems of linear equations. It provides a structured and concise way to capture all the information from the equations. In matrix form, the coefficients of the variables and the constants on the right are arranged neatly in rows and columns.For the given exercise, each equation is translated into a row in the matrix, with each of the variables' coefficients occupying separate columns. This matrix is known as the **augmented matrix** because it includes both the coefficient matrix and an additional column for the constants from the equations.Consider the matrix:

• First row corresponds to the first equation, \(2x + 3y - z = 8\).
• Second row corresponds to the second equation, \(0x + y + 2z = -10\).
• Third row comes from the third equation, \(x - 2y - 3z = 21\).

Arranging them in matrix form helps visualize and solve the system efficiently, often via methods such as Gaussian elimination or by leveraging computational tools.

Linear Algebra

Linear algebra is the branch of mathematics that deals with vectors, vector spaces, and the operations and transformations on them. It also examines linear mappings between these spaces, which is exactly what happens when dealing with systems of linear equations. In the context of our system, linear algebra provides the theoretical foundation for understanding matrix operations. This includes the resolution of these systems through various techniques, such as row operations or matrix factorization, to achieve what is known as row-echelon form or reduced row-echelon form, crucial steps in determining the solutions. These techniques allow for analysis of system properties like consistency, uniqueness of solutions, and more. Hence, understanding linear algebra concepts is critical for mastering how to handle simultaneous equations and their matrix representations effectively.

Coefficient Alignment

Coefficient alignment refers to the careful arrangement of the terms in each linear equation so they align with corresponding terms in other equations. This alignment is crucial when forming the augmented matrix, as it ensures that each column represents a unique variable across the system.For instance, in our system, we ensure that all equations are written with terms corresponding to \(x\), \(y\), and \(z\) aligned vertically:

• Equation 1: \(2x + 3y - z = 8\)
• Equation 2: \(0x + y + 2z = -10\)
• Equation 3: \(x - 2y - 3z = 21\)

Aligning coefficients this way simplifies constructing the augmented matrix, facilitating straightforward application of matrix operations. It ensures that during any manipulations to solve the system, like row reductions, the integrity of the relationships between variables is maintained perfectly.