Problem 18 - A Graphical Approach to College Algebra

Question

Write the system of equations associated with each augmented matrix. $$\left[\begin{array}{lll|l} 1 & 0 & 1 & 4 \\ 0 & 1 & 0 & 2 \\ 0 & 0 & 1 & 3 \end{array}\right]$$

Step-by-Step Solution

Answer

The system is: \(x + z = 4\), \(y = 2\), \(z = 3\).

Step 1: Identify Rows and Variables

The given augmented matrix has three rows, each corresponding to a linear equation. Let's denote the variables as \(x\), \(y\), and \(z\). The columns before the separator '|', represent the coefficients of these variables.

Step 2: Extract First Equation

The first row is \([1, 0, 1 | 4]\). Converting this to an equation, we have \(1 \cdot x + 0 \cdot y + 1 \cdot z = 4\). Simplifying, we get the first equation: \(x + z = 4\).

Step 3: Extract Second Equation

The second row is \([0, 1, 0 | 2]\). Converting this to an equation, we have \(0 \cdot x + 1 \cdot y + 0 \cdot z = 2\). Simplifying, we get the second equation: \(y = 2\).

Step 4: Extract Third Equation

The third row is \([0, 0, 1 | 3]\). Converting this to an equation, we have \(0 \cdot x + 0 \cdot y + 1 \cdot z = 3\). Simplifying, we get the third equation: \(z = 3\).

Step 5: Write the System of Equations

Based on the extracted equations from the previous steps, the system of equations is: \[ \begin{align*} x + z &= 4 \ y &= 2 \ z &= 3 \end{align*} \]

Key Concepts

System of Equations

A system of equations is a collection of two or more equations with a common set of variables. In the world of mathematics, solving a system of equations means finding the values of the variables that satisfy all given equations simultaneously. For example, imagine you have a system consisting of three equations like the one in this exercise:

• \(x + z = 4\)
• \(y = 2\)
• \(z = 3\)

To solve it, you search for values of \(x\), \(y\), and \(z\) that make all these statements true at the same time. In this case, solving the system is relatively straightforward due to each equation being quite simple and mostly involving only one variable. Understanding such systems is crucial as they can represent real-world problems in fields like physics, engineering, and economics.

Linear Equations

Linear equations are a key component in many mathematical problems, especially when dealing with systems of equations. A linear equation is an equation that makes a straight line when it is graphed. These equations don’t involve any powers or roots on the variables. They are always in the form:\[ax + by + cz + ... = d\]where \(a\), \(b\), \(c\) are coefficients, and \(d\) is a constant. In our exercise, the equations \(x + z = 4\), \(y = 2\), and \(z = 3\) are all examples of linear equations. Each one can be represented as a flat line in a multidimensional coordinate space. Linear equations are valuable because they present relationships that are easy to analyze and solve. When multiple linear equations are presented together, we can employ various methods, such as substitution, elimination, or matrix representation, to find solutions.

Matrix Representation

Matrix representation is a powerful tool used to simplify and solve system of equations. It involves writing the coefficients of the variables and the constants from the equations into a matrix form. This makes it easier to manipulate and solve complex systems. An augmented matrix, like the one in this exercise, combines the coefficient matrix with the constant terms from the equations. The vertical bar \(|\) separates the coefficients from the constants. In this exercise, the augmented matrix:\[\begin{array}{ccc|c}1 & 0 & 1 & 4 \0 & 1 & 0 & 2 \0 & 0 & 1 & 3\end{array}\]encapsulates the system of equations such that each row corresponds to an equation. This representation allows us to efficiently utilize mathematical tools, like Gaussian elimination or determinants, to find solutions. Understanding matrix representation is indispensable for higher level mathematics and practical applications in various scientific disciplines.

Variables Identification

Identifying variables is the initial and crucial step in solving equations or systems of equations. Variables are symbols, often letters, used to represent unknown quantities. In the context of the given exercise, we have chosen the variables as \(x\), \(y\), and \(z\). Recognizing which variables are involved and what they represent will guide you through translating an abstract matrix or set of conditions into explicit mathematical equations. This is crucial because each variable acts as a placeholder for a number that can be calculated to satisfy the equation's conditions. Correctly identifying and assigning these placeholders helps in understanding and structuring the problem accurately, ensuring the correct application of mathematical operations when solving the system.