Problem 14 - College Algebra with Corequisite Support

Question

For the following exercises, write the linear system from the augmented matrix. $$ \left[\begin{array}{rrr|r} 8 & 29 & 1 & 43 \\ -1 & 7 & 5 & 38 \\ 0 & 0 & 3 & 10 \end{array}\right] $$

Step-by-Step Solution

Answer

The system is: \(8x + 29y + z = 43\), \(-x + 7y + 5z = 38\), \(3z = 10\).

Step 1: Identify the Variables

In a linear system, each column before the vertical bar represents a coefficient for a variable. Assign variables to these columns, such as \(x\), \(y\), and \(z\).

Step 2: Write the System of Equations

Based on the augmented matrix, write the system of equations by matching the coefficients with the variables. The equations will form as follows:1. First row: \(8x + 29y + 1z = 43\).2. Second row: \(-1x + 7y + 5z = 38\).3. Third row: \(0x + 0y + 3z = 10\).

Step 3: Verify the System

Ensure that each equation matches with its respective row in the augmented matrix and that all coefficients and constants are correctly translated into the equation format.

Key Concepts

Augmented Matrix

An augmented matrix is a simple and organized way of representing a system of linear equations in matrix form. It's essentially a matrix that contains both the coefficients of the variables and the constants from the equations, all in one neat table-like structure. For example, the given matrix for this exercise is:\[\left[\begin{array}{rrr|r} 8 & 29 & 1 & 43 \-1 & 7 & 5 & 38 \0 & 0 & 3 & 10 \end{array}\right] \]Here's how it works:

• The vertical bar separates the coefficients from the constants; coefficients are to the left and constants to the right.
• Each row corresponds to a different equation in the system.

Understanding an augmented matrix helps in simplifying the process of solving linear equations, as it makes it easier to apply techniques like row reduction or Gaussian elimination.

System of Equations

A system of equations is simply a set of equations with multiple variables that you solve together because they're related. In our example, the system of equations we derive from the augmented matrix is:

• Equation 1: \(8x + 29y + 1z = 43\)
• Equation 2: \(-1x + 7y + 5z = 38\)
• Equation 3: \(0x + 0y + 3z = 10\)

Each equation represents a condition that involves the same set of variables (\(x\), \(y\), and \(z\)). Solving this system means finding values for each variable that satisfy all equations at the same time.
Typically, solutions can be found graphically by finding where the equations intersect in space, or algebraically by manipulating the equations to find variable values. Being comfortable writing and interpreting systems of equations from different forms, such as augmented matrices, is essential in linear algebra.

Variables and Coefficients

In the context of linear systems, variables and coefficients are fundamental concepts to grasp. The variables, often denoted as \(x\), \(y\), \(z\), etc., represent the unknowns in the equations that we need to solve for.

• Variables: The variables in our problem, \(x\), \(y\), and \(z\), are the elements we're solving for in the system of equations derived from the matrix.
• Coefficients: These are the numbers in front of the variables, indicating the variable's contribution to each equation. For instance, in the equation \(8x + 29y + 1z = 43\), 8, 29, and 1 are the coefficients for \(x\), \(y\), and \(z\) respectively.

Understanding how coefficients work is important because they determine the weight or significance of each variable within an equation.
When constructing and solving a system of equations from an augmented matrix, the ability to correctly identify and interpret variables and coefficients is integral to achieving the correct solutions.