Consider The Following Three Systems Of Linear Equations

Consider the Following Three Systems of Linear Equations: A Deep Dive into Solutions and Techniques

Linear equations are the backbone of many mathematical models, used to describe relationships between variables in various fields like physics, engineering, economics, and computer science. Understanding how to solve systems of these equations is crucial. This article delves into the intricacies of solving three distinct systems of linear equations, exploring different methods and highlighting the nuances of each approach. We'll examine the concepts of consistency, inconsistency, and uniqueness of solutions, providing a comprehensive understanding of this fundamental mathematical topic.

System 1: A Simple 2x2 System

Let's start with a straightforward example:

2x + y = 5

x - y = 1

This system represents two lines in a Cartesian coordinate system. The solution to the system is the point (if it exists) where these two lines intersect. We can solve this system using several methods:

1. Elimination Method

This method involves manipulating the equations to eliminate one variable. In this case, we can add the two equations together:

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

This simplifies to:

3x = 6

Solving for x, we get:

x = 2

Substituting this value of x back into either of the original equations (let's use the first one):

2(2) + y = 5

Solving for y:

y = 1

Therefore, the solution to this system is (2, 1). This means the two lines intersect at the point (2, 1).

2. Substitution Method

Alternatively, we can use the substitution method. Solve one equation for one variable and substitute it into the other equation. Let's solve the second equation for x:

x = y + 1

Now substitute this expression for x into the first equation:

2(y + 1) + y = 5

Simplifying and solving for y:

2y + 2 + y = 5

3y = 3

y = 1

Substitute this value of y back into either original equation (or the expression for x we derived earlier) to find x:

x = 1 + 1

x = 2

Again, the solution is (2, 1).

Consistency and Uniqueness

This system is consistent, meaning it has at least one solution. It's also unique, meaning it has only one solution. This is visually represented by the two lines intersecting at a single point.

System 2: A 3x3 System with a Unique Solution

Now let's consider a more complex system:

x + y + z = 6

2x - y + z = 3

x + 2y - z = 3

This system has three equations and three unknowns. We'll employ the elimination method to solve it:

1. Eliminate z: Add the first and third equations:

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

This simplifies to:

2x + 3y = 9

2. Eliminate z again: Subtract the second equation from the first:

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

This simplifies to:

-x + 2y = 3

3. Solve for x and y: Now we have a 2x2 system:

2x + 3y = 9

-x + 2y = 3

Multiply the second equation by 2:

-2x + 4y = 6

Add this to the first equation:

(2x + 3y) + (-2x + 4y) = 9 + 6

7y = 15

y = 15/7

Substitute this value of y back into -x + 2y = 3:

-x + 2(15/7) = 3

-x = 3 - 30/7 = -9/7

x = 9/7

Finally, substitute the values of x and y into any of the original equations (let's use the first one) to solve for z:

(9/7) + (15/7) + z = 6

24/7 + z = 6

z = 6 - 24/7 = 18/7

z = 18/7

The solution to this system is (9/7, 15/7, 18/7). This system is also consistent and unique.

System 3: An Inconsistent System

Consider this system:

x + y = 2

2x + 2y = 5

If we attempt to solve this using elimination, multiply the first equation by 2:

2x + 2y = 4

Now compare this to the second equation:

2x + 2y = 5

This is a contradiction: 2x + 2y cannot simultaneously equal 4 and 5. This system is inconsistent, meaning it has no solution. Geometrically, these two equations represent parallel lines that never intersect.

Gaussian Elimination: A Powerful Technique for Larger Systems

For larger systems of linear equations (more than three variables), Gaussian elimination (also known as row reduction) becomes an invaluable tool. This method involves systematically manipulating the equations (using elementary row operations) to transform the system into row-echelon form or reduced row-echelon form. This form makes it straightforward to solve for the variables. The process involves:

1. Row Swapping: Interchange any two rows.
2. Row Multiplication: Multiply a row by a non-zero constant.
3. Row Addition: Add a multiple of one row to another row.

The goal is to create a triangular matrix where the leading coefficient of each row is 1 and the elements below the leading coefficient are zero. This allows for back-substitution to solve for the variables.

Matrix Representation and Determinants

Systems of linear equations can be efficiently represented using matrices. For example, the 3x3 system above can be written as:

[ 1  1  1 ] [ x ]   [ 6 ]
[ 2 -1  1 ] [ y ] = [ 3 ]
[ 1  2 -1 ] [ z ]   [ 3 ]

The determinant of the coefficient matrix plays a crucial role in determining the nature of the solution. A non-zero determinant indicates a unique solution (consistent and unique system), while a zero determinant suggests either no solution (inconsistent system) or infinitely many solutions (consistent dependent system).

Applications of Linear Equations

The applications of linear equations are vast and varied:

• Engineering: Structural analysis, circuit analysis, and control systems.
• Physics: Classical mechanics, electromagnetism, and quantum mechanics.
• Economics: Input-output models, linear programming, and econometrics.
• Computer Graphics: Transformations, projections, and rendering.
• Machine Learning: Linear regression, support vector machines, and neural networks.

Conclusion

Solving systems of linear equations is a fundamental skill with broad applications across numerous fields. Understanding the different methods—elimination, substitution, Gaussian elimination, and the role of matrices and determinants—is essential for tackling various mathematical and real-world problems. The concepts of consistency and uniqueness of solutions are critical for interpreting the results and drawing meaningful conclusions from the models represented by these systems. The choice of method depends on the size and complexity of the system, with Gaussian elimination providing a robust and systematic approach for larger systems. This comprehensive understanding empowers individuals to leverage the power of linear algebra in various disciplines.