Problem 10 - Beginning and Intermediate Algebra

Question

Write a system of linear equations in \(x, y,\) and \(z\) represented by each augmented matrix. $$\left[\begin{array}{rrr|r}1 & 4 & -3 & -5 \\\\-1 & 2 & 5 & 8 \\\6 & -2 & -1 & 3\end{array}\right]$$

Step-by-Step Solution

Answer

The system of linear equations represented by the given augmented matrix is: \(x + 4y - 3z = -5\) \(-x + 2y + 5z = 8\) \(6x - 2y - z = 3\)

Step 1: Extract rows from the augmented matrix

We start by extracting the rows from the augmented matrix which will serve as the coefficients and constants of our linear equations: \(\begin{bmatrix}1 & 4 & -3 & | & -5 \\ -1 & 2 & 5 & | & 8\\ 6 & -2 & -1 & | & 3\end{bmatrix}\)

Step 2: Convert the rows into linear equations

Now, we will convert each row of the augmented matrix into a linear equation. For this, we will consider the first three elements in each row as coefficients of x, y, z respectively, and the last element of the row as the constant term: Row 1: \(1x + 4y - 3z = -5\) Row 2: \(-1x + 2y + 5z = 8\) Row 3: \(6x - 2y - 1z = 3\)

Step 3: Write the system of linear equations

Finally, we have the system of linear equations extracted from the augmented matrix: \(x + 4y - 3z = -5\) \(-x + 2y + 5z = 8\) \(6x - 2y - z = 3\)

Key Concepts

Augmented Matrix

An augmented matrix is a simplified way to represent a system of linear equations. Imagine you have several equations, each involving the same variables. By using an augmented matrix, you can capture all the information from those equations in a concise and organized format.

In an augmented matrix, each row corresponds to a linear equation and each column corresponds to a variable or the constant term on the right-hand side of the equation.

• The first group of columns represents the coefficients of the variables.
• The last column contains the constants from each equation.

For example, the matrix in this exercise captures three equations with the variables \(x\), \(y\), and \(z\).

This representation helps efficiently solve systems of equations, particularly when using techniques like Gaussian elimination or matrix operations.

Systems of Equations

A system of equations refers to a set of two or more equations that are meant to be solved together. Each equation within the system has common variables.

In this exercise, we're dealing with a system of three linear equations in three variables: \(x\), \(y\), and \(z\). The goal is to find a set of values for these variables that will satisfy all the equations simultaneously.

• Sometimes the system might have one unique solution.
• In other cases, the system could have infinitely many solutions.
• Or, it might have no solution at all.

Solving a system of equations can uncover relationships between different variables and is a powerful tool for solving real-world problems in science and engineering.

Coefficients and Constants

In the context of linear equations, coefficients and constants play crucial roles in defining the equations. Coefficients are the numbers that multiply your variables, providing the scale or weight of each variable in the equation.
For a linear equation like \(1x + 4y - 3z = -5\):

• The numbers 1, 4, and -3 are the coefficients.
• They define how each variable \(x\), \(y\), and \(z\) contributes to the equation.

On the other hand, the constant is the standalone number on the other side of the equation. In our case, that's -5, 8, and 3 for the respective equations.

This number indicates the result when the variable expressions reach a balance or equilibrium, forming the relationship the equation describes. Combined, coefficients and constants lay out the basic building blocks of algebraic expressions.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operators like plus or minus. In linear equations, these expressions show the relationship between different variables and constants. They lay the groundwork for forming and interpreting equations.

For instance, a term like \(4y\) in the expression is composed of:

• A coefficient (4) indicating how many times to count the variable \(y\).
• A variable \(y\), which can take on different values.

When writing equations from an augmented matrix, each row is converted into these expressions. Algebraic expressions help in visually assembling and understanding the way variables interact and form the relationships specified in linear equations.

Mastering how to manipulate and interpret algebraic expressions is key in solving systems of equations, as they’re the main language through which relationships between variables are communicated.