Problem 14 - A Graphical Approach to College Algebra

Question

Write the augmented matrix for each system. Do not solve the system. $$\begin{aligned} x &=6 \\ y+2 z &=2 \\ x-3 z &=6 \end{aligned}$$

Step-by-Step Solution

Answer

\( \begin{bmatrix} 1 & 0 & 0 & | & 6 \\ 0 & 1 & 2 & | & 2 \\ 1 & 0 & -3 & | & 6 \end{bmatrix} \)

Step 1: Write the System of Equations

Start with the system of equations provided: \[ x = 6 \] \[ y + 2z = 2 \] \[ x - 3z = 6 \] These are the equations we need to represent in an augmented matrix form.

Step 2: Arrange Equations by Variables

Make sure each equation is aligned with respect to the variables \(x, y, z\). Since the first and third equations involve \(x\), align zeros for missing \(y\) and \(z\): \[ x + 0y + 0z = 6 \] \[ 0x + y + 2z = 2 \] \[ x + 0y - 3z = 6 \]

Step 3: Form the Augmented Matrix

Convert the aligned system into an augmented matrix by writing the coefficients of the variables and the constants in a matrix form. This results in: \[ \begin{bmatrix} 1 & 0 & 0 & | & 6 \ 0 & 1 & 2 & | & 2 \ 1 & 0 & -3 & | & 6 \end{bmatrix} \] Assign the coefficients before the variables as matrix entries and the constants after the equal signs as the final column of the augmented matrix.

Key Concepts

Understanding Systems of Equations

Systems of equations are a set or collection of equations that you deal with all together. They often arise in algebra or other areas of mathematics when you have multiple equations with several unknowns. Solving a system means finding values for the variables that simultaneously satisfy all equations.

Here are some key points about systems of equations:

• Each equation gives information about the relationships between the variables.
• The solution to the system is the set of values that satisfies all the equations simultaneously.
• Systems can be solved using different methods such as substitution, elimination, or graphing.

In our exercise, we are given a system consisting of three equations with three variables: \(x\), \(y\), and \(z\). Matrices provide a more efficient way to handle these systems and are particularly useful when dealing with larger systems.

Matrix Representation and Augmentation

Matrix representation is a compact and structured way to visualize systems of equations. Each equation translates to a row within a matrix and each variable gets its own column. Matrices make calculations, like finding solutions to the system of equations, much more manageable in linear algebra.

The augmented matrix is a special type of matrix used to represent a system of equations. It includes both the coefficients of the variables and the constants on the other side of the equal signs, separated by a vertical bar.

• The vertical bar in the augmented matrix distinguishes between coefficients and constants.
• This format is geared toward applying techniques like Gaussian elimination to further solve the system.
• We translate our equations directly into rows of numbers in the augmented matrix, aligning coefficients with their respective variables.

In the example, the aligned equations yield a matrix with constant terms positioned in the last column: \[ \begin{bmatrix} 1 & 0 & 0 & | & 6 \ 0 & 1 & 2 & | & 2 \ 1 & 0 & -3 & | & 6 \end{bmatrix} \]This clear layout is pivotal for the next steps in solving the system.

Breaking Down Algebraic Equations

Algebraic equations form the building blocks of systems of equations. Each equation typically involves variables and constants connected by algebraic operations like addition, subtraction, multiplication, and sometimes division. Understanding how to manipulate these equations is crucial to working with systems.

To break down algebraic equations, consider:

• The position and role of each term (variable or constant) in the equation.
• How combining equations can help in finding variable values.
• Manipulating terms to isolate specific variables, which is essential for solving.

In the given exercise, notice how we derived equations and aligned them by variables. This is vital for forming the augmented matrix, highlighting how algebraic manipulation is key in representing systems in matrix form.