Problem 2 - Algebra and Trigonometry Real Mathematics, Real People

Question

Fill in the blank. A solution of a system of three linear equations in three unknowns can be written as an ____, which has the form \((x, y, z)\).

Step-by-Step Solution

Answer

The solution to a system of three linear equations in three unknowns is referred to as an 'Ordered Triple'.

Step 1: Recall the Structure of Solutions

In general, a solution of a system of N linear equations in N unknowns can be written in a form analogous to an ordered pair for two variables. Instead, in this case, it's an ordered triple \((x, y, z)\).

Step 2: Giving the Structure a Name

From this knowledge we extract the name of such a structure which is called an 'Ordered Triple'.

Key Concepts

Ordered Triple

When you're dealing with a system that includes three linear equations, you're trying to find values for three unknowns. These unknowns are often labeled as \(x\), \(y\), and \(z\). An ordered triple is simply the solution expressed in the form \((x, y, z)\). This compact notation tells us that \(x\) corresponds to the solution of the first equation, \(y\) to the second, and \(z\) to the third.
An ordered triple is very similar to an ordered pair, which you might be more familiar with, from working with two equations and two unknowns. However, with three values, it needs three coordinates to show the exact point of intersection in 3-dimensional space.
For example, if a system of equations resolves to \((2, -1, 5)\), this means that substituting \(x = 2\), \(y = -1\), and \(z = 5\) into each equation will satisfy all three equations at once.

System of Equations

A system of equations is a set of two or more equations that you work with all at once. In our case, we're focusing on systems with three equations involving three unknowns. Solving such a system means finding the values for the unknowns that satisfy all the equations in the system at the same time.
This approach is essential in problems where multiple conditions need to be fulfilled simultaneously. The system can represent different scenarios like the intersection of planes in geometry or balancing equations in chemistry.

• The solutions can be visualized as points where graphs intersect - in 3D, this intersection can happen at one point, along a line, or over an entire plane.
• When solving these systems, methods such as substitution, elimination, or matrices (using Gaussian elimination) are commonly employed.

The outcome of solving a system of equations in three dimensions is often referred to as finding an ordered triple, which represents a unique solution of the whole system.

Three Unknowns

In equations with three unknowns, you have variables often represented as \(x\), \(y\), and \(z\). They are the values that we aim to determine when we solve the system of equations.
This scenario usually represents a more realistic situation in the real world where three different conditions need to be satisfied simultaneously.
To solve systems with three unknowns, you'll often rely on methods like:

• Substitution: Solve one equation for one unknown, then substitute that solution into the other equations.
• Elimination: Add or subtract equations to eliminate variables and solve step-by-step.
• Matrix methods: Use mathematical structures that help to simplify complex systems into manageable calculations.

Mastering techniques for solving systems with three unknowns allows for a better understanding of multi-variable problems and improves problem-solving skills across various mathematical applications.