A System Of Linear Equations With Exactly One Solution

A System of Linear Equations with Exactly One Solution: A Comprehensive Guide

A system of linear equations is a collection of two or more linear equations with the same set of variables. Finding the solution, or solutions, to this system involves determining the values of the variables that satisfy all equations simultaneously. One crucial aspect of understanding linear systems is identifying when a system possesses exactly one solution. This article will delve deep into the characteristics, methods of solving, and real-world applications of such systems.

Understanding Linear Systems and Their Solutions

Before we focus on systems with exactly one solution, let's establish a broader understanding. A system of linear equations can have one of three possible outcomes:

• Exactly one solution: The system has a unique set of values for the variables that satisfy all equations. This is the focus of this article.
• Infinitely many solutions: The equations are linearly dependent, meaning one equation is a multiple of another. This results in an infinite number of solutions satisfying the system.
• No solution: The equations are inconsistent, meaning there are no values for the variables that satisfy all equations simultaneously. The lines (or planes in higher dimensions) representing the equations are parallel.

Identifying Systems with Exactly One Solution

A system of linear equations with exactly one solution is characterized by its independent equations. This independence means that no equation can be obtained by a linear combination of the other equations. Geometrically, in two dimensions, this translates to two lines intersecting at a single point. In three dimensions, it means three planes intersecting at a single point.

Let's examine a simple example:

x + y = 5
x - y = 1

This system has exactly one solution because the lines representing these equations have different slopes and thus intersect at a unique point. We can solve this system using various methods (detailed below) to find x = 3 and y = 2.

Methods for Solving Systems with Exactly One Solution

Several robust methods exist for solving systems of linear equations with exactly one solution. The choice of method often depends on the number of equations and variables, as well as the complexity of the coefficients.

1. Substitution Method

The substitution method involves solving one equation for one variable and substituting that expression into the other equations. This process is repeated until a single equation with one variable remains, which can then be solved. This solution is then substituted back into previous equations to find the values of the other variables.

Example: Let's revisit the system above:

x + y = 5  (Equation 1)
x - y = 1  (Equation 2)

From Equation 2, we can solve for x: x = y + 1

Substituting this into Equation 1:

(y + 1) + y = 5

Solving for y: 2y = 4, so y = 2

Substituting y = 2 back into x = y + 1, we get x = 3.

Therefore, the solution is x = 3, y = 2.

2. Elimination Method (or Gaussian Elimination)

The elimination method focuses on eliminating variables by adding or subtracting multiples of equations. The goal is to reduce the system to a single equation with one variable.

Example: Using the same system:

x + y = 5  (Equation 1)
x - y = 1  (Equation 2)

Adding Equation 1 and Equation 2 eliminates y:

(x + y) + (x - y) = 5 + 1 2x = 6 x = 3

Substituting x = 3 into either Equation 1 or Equation 2 gives y = 2.

3. Matrix Methods (Gaussian Elimination and Gauss-Jordan Elimination)

For larger systems, matrix methods are highly efficient. These methods represent the system of equations as an augmented matrix and use row operations to transform the matrix into row-echelon form (Gaussian elimination) or reduced row-echelon form (Gauss-Jordan elimination). The solution is then readily apparent from the transformed matrix.

Example: The system:

x + y = 5
x - y = 1

Can be represented as the augmented matrix:

[ 1  1 | 5 ]
[ 1 -1 | 1 ]

Through row operations (e.g., subtracting row 1 from row 2), we can transform this matrix into row-echelon form, revealing the solution.

4. Cramer's Rule

Cramer's rule is a direct method for solving systems of linear equations using determinants. It's particularly useful for smaller systems (2x2 or 3x3) but becomes computationally expensive for larger systems.

Example: For a 2x2 system:

ax + by = c
dx + ey = f

The solution can be found using:

x = (ce - bf) / (ae - bd) y = (af - cd) / (ae - bd)

provided that ae - bd ≠ 0 (ensuring a unique solution).

Real-World Applications of Systems with Exactly One Solution

Systems of linear equations with exactly one solution are fundamental to numerous real-world applications across various disciplines:

• Engineering: Analyzing circuits, determining forces in structural systems, and solving heat transfer problems. Each component or element contributes a linear equation, and finding the solution provides the unique values of currents, forces, or temperatures within the system.

• Economics: Modeling supply and demand, optimizing production, and analyzing market equilibrium. The intersection of supply and demand curves represents the unique equilibrium price and quantity.

• Computer Graphics: Transforming objects in 3D space, rendering images, and performing simulations. The calculations involved often rely on solving systems of linear equations to accurately position and represent objects.

• Physics: Solving problems related to Newtonian mechanics, electromagnetism, and fluid dynamics. These fields frequently involve sets of linear equations that, when solved, provide unique values for variables like velocity, acceleration, or electric field strength.

• Data Science and Machine Learning: Linear regression, a cornerstone of statistical modeling, involves finding the best-fitting line through data points. This involves solving a system of linear equations to determine the optimal parameters of the linear model.

Checking Your Solution

After obtaining a solution, it's crucial to verify its correctness by substituting the values back into the original equations. If the solution satisfies all equations, it is indeed the unique solution to the system. This verification step is essential to ensure the accuracy of your calculations and avoid errors.

Conclusion

Systems of linear equations with exactly one solution form a fundamental concept in mathematics and possess wide-ranging applications in diverse fields. Understanding the characteristics of such systems, mastering different solution methods, and recognizing their real-world significance is essential for anyone working with mathematical modeling and problem-solving. The choice of method depends on the complexity of the system, and verification is always a crucial final step to ensure accuracy. By grasping these concepts, you’ll gain valuable tools for tackling a wide array of quantitative challenges.