Problem 15 - College Algebra

Question

For the following exercises, write the linear system from the augmented matrix. $$ \left[\begin{array}{ccc|c}{4} & {5} & {-2} & {12} \\ {0} & {1} & {58} & {2} \\\ {8} & {7} & {-3} & {-5}\end{array}\right] $$

Step-by-Step Solution

Answer

The linear system is: \(4x + 5y - 2z = 12\), \(y + 58z = 2\), \(8x + 7y - 3z = -5\).

Step 1: Understand the augmented matrix

The given matrix is augmented, meaning it combines the coefficients of variables and the constants from the equations on the right side of the vertical bar. The numbers to the left of the vertical bar represent the coefficients of each variable in the linear system.

Step 2: Identify variables for the linear system

Assign variables to each column in the coefficient section. Let's denote the first column for variable \(x\), the second for \(y\), and the third for \(z\). Thus, each row of the coefficients before the bar will correspond to the variables \(x, y,\) and \(z\).

Step 3: Write the first equation from the first row

The first row of the matrix is \([4, 5, -2 | 12]\). Using the assigned variables, the equation becomes:\[4x + 5y - 2z = 12\]

Step 4: Write the second equation from the second row

The second row of the matrix is \([0, 1, 58 | 2]\). This translates to the equation:\[0x + 1y + 58z = 2\]Simplified, this is:\[y + 58z = 2\]

Step 5: Write the third equation from the third row

The third row of the matrix is \([8, 7, -3 | -5]\). Using the assigned variables, the equation becomes:\[8x + 7y - 3z = -5\]

Key Concepts

Understanding Linear Systems

A linear system is a collection of equations that involve the same set of variables. Each equation in the system represents a straight line when graphed, and the solution to the system is the point or set of points where these lines intersect. Linear systems can have:

• One unique solution: All lines intersect at a single point.
• Infinitely many solutions: The lines overlap entirely.
• No solution: The lines are parallel and never intersect.

In the context of a matrix, a linear system is often conveniently expressed in terms of rows and columns. Each row of the matrix corresponds to one equation in the system.

Role of Coefficients

Coefficients are the numerical factors that precede the variables in equations. They play a crucial role in defining the slope and position of each line or plane represented by the equations. For instance, in the equation \(4x + 5y - 2z = 12\) from the matrix, the numbers 4, 5, and -2 are the coefficients of the variables \(x\), \(y\), and \(z\) respectively.
In a matrix form, coefficients are systematically organized into rows:

• The first row contains coefficients for the first equation.
• The second row contains coefficients for the second equation.
• And so on for any additional rows.

In an augmented matrix, these coefficients are to the left of the vertical line, making them easy to distinguish from the constants or solutions on the right.

Breaking Down Equations

Equations are mathematical statements that show the relationship between different variables and constants through equality signs. In a linear system, each line of the augmented matrix represents an equation. For example, the first row \([4, 5, -2 | 12]\) translates to the equation \(4x + 5y - 2z = 12\).
Each equation in the matrix form follows the general layout:

• Coefficients multiply their respective variables.
• The constants or resulting values follow after the vertical line.

These equations can then be manipulated using algebraic methods, such as substitution or elimination, to find the values of the variables.

Unpacking Variables

Variables are symbols that represent unknown values in equations. In a linear system expressed through an augmented matrix, variables like \(x\), \(y\), and \(z\) are placeholders for actual numbers that solve the system. Assigning variables systematically across the columns allows us to rewrite the matrix into a recognizable system of equations.
Commonly, variables are labeled:

• \(x\), \(y\), \(z\), etc., corresponding to each column of the matrix before the vertical line.
• This ensures clarity and consistency when converting between the matrix and its equivalent system of equations.

By solving the linear system, we determine the values of these variables that satisfy each equation simultaneously.