A System Of Linear Equations With Exactly One Solution

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

Understanding systems of linear equations is fundamental in mathematics, with applications spanning numerous fields, from engineering and computer science to economics and finance. This article delves into systems of linear equations that possess exactly one solution – a crucial concept often encountered in various mathematical contexts. We'll explore the characteristics of such systems, methods for solving them, and the underlying geometrical interpretations. This guide aims to provide a thorough understanding, suitable for students and anyone interested in deepening their mathematical knowledge.

Introduction: What are Systems of Linear Equations?

A system of linear equations is a collection of two or more linear equations involving the same set of variables. A linear equation is an algebraic equation of the form ax + by + cz + ... = k, where 'a', 'b', 'c', and 'k' are constants, and 'x', 'y', 'z' are variables. The goal is to find values for the variables that satisfy all equations simultaneously. These solutions represent points of intersection between the lines (or planes, in higher dimensions) represented by the equations.

A system can have:

• Exactly one solution: The lines (or planes) intersect at a single point.
• Infinitely many solutions: The lines (or planes) coincide, overlapping completely.
• No solution: The lines (or planes) are parallel and never intersect.

This article focuses exclusively on systems with exactly one solution, exploring their properties and solution methods.

Characteristics of Systems with Exactly One Solution

Systems with exactly one solution are characterized by specific relationships between their equations. Crucially, no equation can be a multiple of another, implying linear independence. This means that no equation can be obtained by multiplying another equation by a constant. Geometrically, this translates to lines (in two dimensions) or planes (in three dimensions) that are not parallel and intersect at a unique point.

Consider a simple example in two dimensions:

x + y = 3 x - y = 1

These two equations represent two distinct lines that are not parallel. Their intersection point represents the unique solution to the system. We can solve this system using various methods, as described below.

Methods for Solving Systems with Exactly One Solution

Several robust methods exist for finding the unique solution to a system of linear equations. We'll discuss three prominent techniques:

1. Elimination Method (or Gaussian Elimination):

This method involves systematically eliminating variables by adding or subtracting multiples of equations until a single equation with a single variable is obtained. This process is then repeated to solve for the remaining variables.

Let's use the elimination method on our example:

x + y = 3 (Equation 1) x - y = 1 (Equation 2)

Adding Equation 1 and Equation 2 eliminates 'y':

2x = 4 x = 2

Substituting x = 2 into Equation 1:

2 + y = 3 y = 1

Therefore, the unique solution is x = 2 and y = 1.

This method can be extended to larger systems involving more variables, often organized using matrices to streamline the calculations.

2. Substitution Method:

This method involves solving one equation for one variable in terms of the other variables and then substituting this expression into the remaining equations. This reduces the number of variables and simplifies the system.

Using the same example:

x + y = 3 (Equation 1) x - y = 1 (Equation 2)

Solve Equation 1 for 'x':

x = 3 - y

Substitute this expression for 'x' into Equation 2:

(3 - y) - y = 1 3 - 2y = 1 2y = 2 y = 1

Substitute y = 1 back into x = 3 - y:

x = 3 - 1 x = 2

Again, the unique solution is x = 2 and y = 1. This method is particularly useful for smaller systems.

3. Matrix Methods (Gaussian Elimination & Inverse Matrices):

For larger systems, matrix methods offer a more efficient and organized approach. The system of equations can be represented in matrix form as AX = B, where 'A' is the coefficient matrix, 'X' is the column vector of variables, and 'B' is the column vector of constants.

Gaussian elimination can be performed on the augmented matrix [A|B] to obtain row-echelon form, allowing for back-substitution to find the solution. Alternatively, if matrix 'A' is invertible (meaning its determinant is non-zero – a condition ensuring a unique solution), the solution can be found directly using the inverse matrix: X = A⁻¹B.

This method requires knowledge of matrix algebra and operations such as finding determinants and inverses. Software tools and calculators are frequently used for solving larger systems using matrix methods.

Geometrical Interpretation

The geometrical interpretation of systems of linear equations provides valuable insights.

• Two variables (2D): Each linear equation represents a line in the Cartesian plane. A system with exactly one solution means the lines intersect at a single point, the coordinates of which represent the solution. Parallel lines represent a system with no solution, while coincident lines indicate infinitely many solutions.

• Three variables (3D): Each linear equation represents a plane in three-dimensional space. A system with exactly one solution corresponds to three planes intersecting at a single point. Various configurations of intersecting and parallel planes can lead to systems with no solution or infinitely many solutions.

The Role of the Determinant

The determinant of the coefficient matrix plays a significant role in determining the nature of the solution. For a system of 'n' linear equations with 'n' variables, the determinant of the coefficient matrix 'A' (denoted as det(A) or |A|) provides crucial information:

• det(A) ≠ 0: The system has exactly one solution. The lines (or planes) intersect at a single point. The matrix 'A' is invertible.

• det(A) = 0: The system either has no solution or infinitely many solutions. The lines (or planes) are either parallel or coincident.

Examples with Higher Dimensions

The principles discussed above extend to systems with more than two or three variables. Consider a system of three equations with three unknowns:

x + y + z = 6 x - y + z = 2 2x + y - z = 3

This system can be solved using any of the methods described previously, resulting in a unique solution if the determinant of the coefficient matrix is non-zero. The geometrical interpretation involves the intersection of three planes in three-dimensional space.

Applications of Systems with Exactly One Solution

Systems of linear equations with exactly one solution find extensive applications in various fields:

• Engineering: Analyzing circuits, structural mechanics, and fluid dynamics often involve solving systems of linear equations.
• Computer Science: Computer graphics, cryptography, and machine learning algorithms rely heavily on linear algebra and the solution of linear systems.
• Economics: Input-output analysis, linear programming, and econometric modeling frequently utilize systems of linear equations.
• Physics: Solving problems in mechanics, electromagnetism, and quantum mechanics often requires finding solutions to systems of linear equations.

Frequently Asked Questions (FAQ)

Q1: What if the determinant is zero?

A1: If the determinant of the coefficient matrix is zero, the system either has no solution (inconsistent) or infinitely many solutions (consistent and dependent). Further analysis is needed to determine which case applies. Methods like Gaussian elimination can reveal the nature of the solution in these scenarios.

Q2: Can I always use any method to solve a system?

A2: While all the methods described can be applied to systems with exactly one solution, the choice of method often depends on the size and complexity of the system. For smaller systems, substitution or elimination might be easier. For larger systems, matrix methods are often more efficient and organized.

Q3: How can I check my solution?

A3: Once you have obtained a potential solution, substitute the values of the variables back into the original equations. If the equations are all satisfied, the solution is correct.

Q4: What if I have more equations than unknowns?

A4: This represents an overdetermined system. Generally, such systems will not have a solution unless there are redundant equations.

Q5: What if I have fewer equations than unknowns?

A5: This is an underdetermined system, and typically will have infinitely many solutions.

Conclusion

Systems of linear equations with exactly one solution are a cornerstone of mathematics and its applications. Understanding their characteristics, the available solution methods (elimination, substitution, matrix methods), and the geometrical interpretations are essential for success in various fields. The determinant of the coefficient matrix serves as a key indicator of the nature of the solution. While this guide provides a comprehensive overview, further exploration into linear algebra will enhance your understanding and ability to solve more complex systems. Mastering these concepts opens doors to a deeper appreciation of mathematics and its vast applications in the real world.