Problem 12 - College Algebra

Question

write the system of linear equations represented by the augmented matrix. Use \(x, y,\) and \(z,\) or, if necessary, \(w, x, y,\) and \(z,\) for the variables. $$ \left[\begin{array}{rrrr|r} 4 & 1 & 5 & 1 & 6 \\ 1 & -1 & 0 & -1 & 8 \\ 3 & 0 & 0 & 7 & 4 \\ 0 & 0 & 11 & 5 & 3 \end{array}\right] $$

Step-by-Step Solution

Answer

The system of linear equations represented by the augmented matrix is: \n 4w + x + 5y + z = 6, \n w - x - z = 8, \n 3w + 7z = 4, \n 11y + 5z = 3

Step 1: Assign Variables

Assign a variable for each column. From left to right, the variables will be \(w, x, y,\) and \(z.\) The rightmost column are the results of the equations.

Step 2: Formulate Equations

Convert each row into an equation. The coefficients in front of the variables are given by the numbers in each cell of the row. It will look as follows: \n Row 1: 4w + x + 5y + z = 6 \n Row 2: w - x - z = 8 \n Row 3: 3w + 7z = 4 \n Row 4: 11y + 5z = 3

Key Concepts

Understanding a System of Linear Equations

A system of linear equations is a collection of two or more linear equations that involve the same set of variables. These systems can typically be solved using methods such as substitution, elimination, or matrix operations. When you have a system, each equation represents a line or a plane, and the solution consists of the values of the variables that simultaneously satisfy all the equations in the system.
It is particularly useful in various fields such as physics, engineering, and economics where relationships between variables need to be analyzed and understood.
In our exercise, we work with a system involving four variables, which is associated with a matrix that helps simplify the process of solving.

Variables Assignment in Matrix Representation

In the context of an augmented matrix, each column (except the last) corresponds to a specific variable in the system of equations. Assigning variables in such a way simplifies the transformation from matrix to equation form.
For each column, we assign variables as follows:

• The first column is associated with the variable \( w \).
• The second column aligns with \( x \).
• The third column relates to \( y \).
• The fourth column corresponds to \( z \).

The last column of the matrix represents the constants from the right-hand side of the equations. This labeling facilitates the conversion from the matrix format back into a system of equations.

Matrix to Equations Conversion

The conversion from an augmented matrix to a system of equations involves transforming each row of the matrix into its equivalent linear equation. Each number in a row acts as a coefficient for its corresponding variable.
Let's break this down with an example from our exercise:

• Row 1 becomes the equation: \( 4w + x + 5y + z = 6 \). Here, 4, 1, 5, and 1 are coefficients of \( w, x, y, \) and \( z \) respectively, with 6 as the constant on the right-hand side.
• Row 2 is expressed as: \( w - x - z = 8 \), forming another equation.
• Row 3 turns into: \( 3w + 7z = 4 \), emphasizing the idea that an equation may not include all variables, depending on the scenario described by the system.
• Row 4 translates to: \( 11y + 5z = 3 \), again using the left-most numbers as coefficients.

This conversion is vital to extract the system of equations from matrix form.

Interpreting Coefficients in Linear Equations

When dealing with linear equations, coefficients are the numbers placed in front of variables. They signify how much a change in one variable affects the entire equation and thus represent the weight/spread of that variable within the equation.
In our exercise, initial placement of numbers from the augmented matrix informs us about the influence and relationship of each variable in an equation. For example:

• In \( 4w + x + 5y + z = 6 \), the coefficient 4 signifies the relationship and impact of \( w \) in the equation.
• Negative coefficients, such as \( -x \) or \( -z \) in \( w - x - z = 8 \), denote an inverse relationship, meaning as the variable increases, the overall equation’s value decreases.
• Some variables might not appear in every equation, as shown in \( 3w + 7z = 4 \), where \( x \) and \( y \) have coefficients of zero indicating they don’t influence that specific equation.

Understanding these coefficients helps reveal how different variables interact and impact each other within a system.