Problem 17 - A Graphical Approach to College Algebra

Question

Write the system of equations associated with each augmented matrix. $$\left[\begin{array}{lll|r} 1 & 0 & 0 & 2 \\ 0 & 1 & 0 & 3 \\ 0 & 0 & 1 & -2 \end{array}\right]$$

Step-by-Step Solution

Answer

The system is: \( x = 2 \), \( y = 3 \), \( z = -2 \).

Step 1: Understanding the Augmented Matrix

The given matrix is an augmented matrix representing a system of equations. An augmented matrix includes both the coefficients of the variables and the constants of the equations.

Step 2: Identifying Variables and Equations

The matrix has three rows and three columns before the vertical bar, indicating three variables. We can assume these variables are \( x \), \( y \), and \( z \). Thus, each row corresponds to an equation in these variables.

Step 3: Writing the First Equation

The first row of the matrix is \( [1, 0, 0 | 2] \), which translates to the equation \( 1x + 0y + 0z = 2 \), simplifying to \( x = 2 \).

Step 4: Writing the Second Equation

The second row of the matrix is \( [0, 1, 0 | 3] \), which translates to the equation \( 0x + 1y + 0z = 3 \), simplifying to \( y = 3 \).

Step 5: Writing the Third Equation

The third row of the matrix is \( [0, 0, 1 | -2] \), which translates to the equation \( 0x + 0y + 1z = -2 \), simplifying to \( z = -2 \).

Key Concepts

system of equations

A system of equations consists of two or more equations with the same set of variables. When solving a system, the goal is to find the values of these variables that satisfy all the equations simultaneously. To visualize this better, think of each equation representing a line in two dimensions or a plane in three dimensions. The solution to the system is the point or points where these lines or planes intersect.

For example, the system of equations derived from the given augmented matrix includes three equations:

• Equation 1: \( x = 2 \)
• Equation 2: \( y = 3 \)
• Equation 3: \( z = -2 \)

In this case, since each equation involves only one variable, the solution is straightforward and corresponds to a unique point \((x, y, z) = (2, 3, -2)\). This is an example of a system with a single solution.

matrix representation

Matrices provide a compact and systematic way to represent and solve systems of equations. In a matrix form, each row corresponds to an individual equation and each column corresponds to a particular variable or constant. The coefficients for each variable and the equalities are structured in order to streamline the process of solving the equations.

An augmented matrix, like the one in our example, is a type of matrix that combines the coefficients of the variables and the constants on the right-hand side of the equations. This is done using a vertical bar to separate them.

• The part of the matrix to the left of the vertical bar represents the coefficients of the variables \(x\), \(y\), and \(z\).
• The column to the right of the vertical bar contains the constants from the equations.

This form makes it easier to perform operations like row reduction, especially when using methods such as Gaussian elimination to solve the system.

variables and equations

Variables are symbols used to represent unknown values in equations. In mathematical terms, they serve as placeholders for numbers we want to find when we solve equations or systems of equations. Each equation in the system gives specific information about the relationships between these variables.

In our context, the variables are shown as \(x\), \(y\), and \(z\). Each variable appears in a separate row of the augmented matrix, making it easy to extract individual equations. This method highlights how variables interact in different equations and provides a clear path to find distinct values for each variable given the system's constraints.

• In the equation 1: \(x = 2\), \(x\) is the variable for which the exact value is provided.
• Equation 2, \(y = 3\), shows the value of \(y\) directly.
• Similarly, \(z = -2\) in equation 3 tells us the value of variable \(z\).

Such clarity in representation through variables and equations makes solving more complex systems with multiple interdependent variables feasible.