Problem 10 - College Algebra with Corequisite Support

Question

For the following exercises, write the augmented matrix for the linear system. $$ \begin{array}{r} 6 x+12 y+16 z=4 \\ 19 x-5 y+3 z=-9 \\ x+2 y=-8 \end{array} $$

Step-by-Step Solution

Answer

The augmented matrix is \( \begin{bmatrix} 6 & 12 & 16 & | & 4 \\ 19 & -5 & 3 & | & -9 \\ 1 & 2 & 0 & | & -8 \end{bmatrix} \).

Step 1: Write the Coefficients and Constants

To form an augmented matrix, we first need to write down the coefficients of each variable and the constants from the given system of equations. The equations are: \(6x + 12y + 16z = 4\), \(19x - 5y + 3z = -9\), and \(x + 2y = -8\). Identify the coefficients: for the first equation, coefficients are \(6\), \(12\), \(16\); for the second equation, \(19\), \(-5\), \(3\); for the third equation, \(1\), \(2\), and \(0\) (since there's no \(z\)-term). The constants are \(4\), \(-9\), and \(-8\), respectively.

Step 2: Arrange the Coefficients into Matrix Form

Convert the system of equations into matrix form by arranging the coefficients and constants into the matrix. The augmented matrix includes the coefficients of the variables \(x\), \(y\), \(z\) in the left part and the constants on the right side after a vertical line. The augmented matrix is: \[\begin{bmatrix} 6 & 12 & 16 & | & 4 \ 19 & -5 & 3 & | & -9 \ 1 & 2 & 0 & | & -8 \end{bmatrix}\].

Key Concepts

Linear System

A linear system consists of multiple linear equations that are considered simultaneously. It's called "linear" because each equation represents a straight line when graphed and doesn't involve any powers or products of variables. For example, in the given exercise, you have three equations where the variables, namely \(x\), \(y\), and \(z\), follow a straight-line pattern.

• The main goal when dealing with a linear system is to find the common solution for all equations involved.
• The solution represents a set of values for \(x\), \(y\), and \(z\) that make all individual equations true at the same time.
• This system of equations can specifically have one solution (an intersection point), no solution (parallel lines), or infinitely many solutions (identical lines).

Understanding linear systems is crucial because they're the foundation of numerous real-world problems and advanced mathematical concepts.

Matrix Form

Matrix form is a structured way to organize the information derived from a linear system into a tabular format. It simplifies the process of solving and understanding relationships between equations.
Matrix form breaks down like this:

• The coefficients of each variable are organized into rows and columns based on their positions in the equations.
• This arrangement helps in performing operations like row reduction and using techniques such as Gaussian elimination.
• The given exercise uses an augmented matrix, which means it combines coefficients and constants, allowing for integrated problem-solving approaches.

Writing equations in matrix form is not only a mathematical technique but also a standardized way to deal with complex computations effectively.

Coefficients

Coefficients are the numerical values placed before variables in an equation and they depict a weighted multiple of each variable.
In the context of the exercise:

• The coefficient for each term signals how much of that variable component is used in shaping the equation.
• For instance, in the equation \(6x + 12y + 16z = 4\), the coefficients are \(6\), \(12\), and \(16\) corresponding to \(x\), \(y\), and \(z\) respectively.
• The power of coefficients is pivotal in matrix operation because they form the core structure of the matrix in which we perform calculations.

Without coefficients, it would be impossible to determine the actual lessons and interrelations the equations set out to express, underscoring their importance in linear algebraic systems.

Constants

Constants are the fixed values on the right side of the equations in a linear system, representing the total or sum that each equation must reach.
In mathematical terms:

• They are the numbers not attached to any variables in the equation, such as \(4\), \(-9\), and \(-8\) in the given exercise.
• The constants hold crucial information about the outcome value that the weighted variables (coefficients and variables combined) need to equal.
• In an augmented matrix, constants are typically isolated from the coefficients by a vertical line, clearly distinguishing what is known from what needs to be determined.

By calculating and manipulating these equations governed by constants, mathematicians can evaluate practical problems and situations requiring specific results.