Problem 10 - College Algebra

Question

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

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: Identify the Coefficients

First, identify the coefficients of each variable in the given system of equations. For the first equation \(6x + 12y + 16z = 4\), the coefficients are \(6, 12, 16\), and the constant is \(4\). For the second equation \(19x - 5y + 3z = -9\), the coefficients are \(19, -5, 3\), and the constant is \(-9\). For the third equation \(x + 2y = -8\), the coefficients are \(1, 2, 0\), and the constant is \(-8\).

Step 2: Construct the Augmented Matrix

Create the augmented matrix using the coefficients and constants identified in the previous step. The matrix is written as follows:\[\begin{bmatrix}6 & 12 & 16 & \vline & 4 \19 & -5 & 3 & \vline & -9 \1 & 2 & 0 & \vline & -8\end{bmatrix}\]

Key Concepts

Coefficients

When dealing with a linear system of equations, one crucial concept is the idea of coefficients. These are the numbers that directly multiply the variables in each equation. A coefficient tells us how much a corresponding variable is worth in a certain equation.
In the given system, each equation is composed of variables accompanied by their coefficients. For example, in the equation \(6x + 12y + 16z = 4\), the coefficients are \(6\) for \(x\), \(12\) for \(y\), and \(16\) for \(z\). These coefficients represent how prominently each variable contributes to the equation.

• Coefficient of \(x\): Multiplies the variable \(x\) to form a product that is part of the equation's left side.
• Coefficient of \(y\): Functions similarly for the variable \(y\).
• Coefficient of \(z\): Determines the weight of \(z\) in the equation.

Essentially, understanding coefficients is key to manipulating and working with linear equations effectively.

Linear System

A linear system consists of multiple linear equations that we solve together. Each equation describes a relationship between the same set of variables, and the solution to a linear system is the point or points that satisfy all equations simultaneously.
In simpler terms, a linear system seeks to find common values for variables that work in all given equations. For example, our system is:

• \(6x + 12y + 16z = 4\)
• \(19x - 5y + 3z = -9\)
• \(x + 2y = -8\)

We'll look for values of \(x\), \(y\), and \(z\) that satisfy all three equations at once. A solution to these equations could represent an intersection of surfaces or lines in a multi-dimensional space.
Solving a linear system is all about finding this intersection, where the equations' relationships perfectly align.

Matrix Representation

The matrix representation of a linear system of equations condenses information into a grid-like format. This transformation allows us to handle equations more efficiently through methods like row reduction. By expressing the equations as a matrix, we can perform systematic operations directly on the matrix, instead of cumbersome manipulation of entire equations.
An augmented matrix for the given system is constructed using the coefficients and the constant terms of each equation. This means:
\[\begin{bmatrix}6 & 12 & 16 & \vline & 4 \19 & -5 & 3 & \vline & -9 \1 & 2 & 0 & \vline & -8\end{bmatrix}\]

• The rows of the matrix represent each equation.
• The columns correspond to the coefficients of \(x\), \(y\), \(z\), and their respective constants.

The vertical line in the augmented matrix (\(\vline\)) signifies the separation between the coefficients of the variables and the constants from each equation. Using a matrix simplifies the process of finding solutions, making it an essential tool in solving systems of linear equations.