Problem 3 - Intermediate Algebra

Question

Fill in the blanks. Solutions of a system of three equations in three variables, \(x, y\) and \(z,\) are written in the form \((x, y, z)\) and are called ordered ____.

Step-by-Step Solution

Answer

Solutions are called ordered triples.

Step 1: Understanding the Problem

The problem asks for the term used to describe solutions of a system of equations in three variables in a specific form.

Step 2: Recognizing the Format

We are given that solutions are expressed in the form \((x, y, z)\). This format is a tuple that represents a specific order of values, each corresponding to one of the variables \(x\), \(y\), and \(z\).

Step 3: Identifying the Term

The specific arrangement of numbers in a tuple, where each position is identified with a variable, is known as an 'ordered pair' in two dimensions. In three dimensions, this term is 'ordered triple'.

Key Concepts

ordered triples

When dealing with systems of equations in three variables, the solutions are often represented as an ordered triple. An ordered triple is a set of three numbers, like \(x, y, z\), each representing a solution for one of the variables in the equations: \(x, y\), and \(z\).

The term "ordered" highlights that position matters. In mathematics, the location of each value within the parentheses determines which variable it corresponds to. For example:

• The first position is always for \(x\)
• The second position is for \(y\)
• The third position is for \(z\)

Think of it as coordinates in three-dimensional space, where each set of numbers describes a distinct point in that space. If you were to swap the position of two numbers, you would reference an entirely different point.

three variables

In many mathematical problems, especially those involving systems of equations, we encounter situations where three variables are present. These variables are usually represented by the symbols \(x, y,\) and \(z\).

Having three variables allows for more complex and realistic modeling of problems in three dimensions, much like how they are used to define points in 3D space. In system equations:

• \(x\) can represent one dimension, such as length
• \(y\) can represent width
• \(z\) can represent height or depth

Understanding the role of each variable is crucial because solving these equations requires finding the values for all three variables simultaneously. This is often necessary in fields like physics and engineering, where multidimensional problems are common.

solutions of equations

Solutions to equations are the values that satisfy all of the equations in a system. For systems involving three variables, solutions are found as ordered triples, \(x, y, z\), that fit the criteria set by each equation.

Finding solutions requires solving the system, which can be done using techniques like substitution, elimination, or matrix methods. Each solution must:

• Satisfy every equation in the system simultaneously
• Represent one point of intersection between the equations when graphed
• Be consistent across all equations, meaning it doesn't contradict any within the system

In a real-world context, these solutions can represent practical outcomes, like predicting how different factors might intersect in a scientific model or economic scenario.