Solution Of A System Of Linear Equations Example

Diving into the heart of linear algebra, solving systems of linear equations is a fundamental skill with wide-ranging applications, from engineering and computer science to economics and data analysis. Understanding the methods to tackle these systems, and seeing practical examples, empowers you to solve real-world problems efficiently.

What is a System of Linear Equations?

A system of linear equations is a collection of two or more linear equations involving the same variables. A linear equation is an equation where the highest power of any variable is one. Geometrically, each linear equation in a two-dimensional space represents a straight line. The solution to the system is the point where all the lines intersect. In higher dimensions (three or more variables), each linear equation represents a hyperplane, and the solution is the intersection of all the hyperplanes.

General Form:

A system of m linear equations with n unknowns (variables) can be written in the general form:

a₁₁x₁ + a₁₂x₂ + ... + a₁nxn = b₁ a₂₁x₁ + a₂₂x₂ + ... + a₂nxn = b₂ ... am₁x₁ + am₂x₂ + ... + amnxn = bm

Where:

• x₁, x₂, ..., xn are the unknowns (variables).
• a₁₁, a₁₂, ..., amn are the coefficients of the variables.
• b₁, b₂, ..., bm are the constants (also known as the right-hand side values).

Matrix Representation:

A system of linear equations can be conveniently represented in matrix form as:

Ax = b

Where:

• A is the coefficient matrix (an m x n matrix containing the coefficients aᵢⱼ).
• x is the variable vector (an n x 1 column vector containing the unknowns xᵢ).
• b is the constant vector (an m x 1 column vector containing the constants bᵢ).

Types of Solutions:

A system of linear equations can have three possible types of solutions:

• Unique Solution: The system has exactly one solution, meaning there's one unique set of values for the variables that satisfies all equations. Geometrically, this corresponds to the lines (or hyperplanes) intersecting at a single point.
• No Solution: The system is inconsistent, meaning there is no set of values for the variables that satisfies all equations simultaneously. Geometrically, this means the lines (or hyperplanes) are parallel and never intersect.
• Infinitely Many Solutions: The system is dependent, meaning there are infinitely many sets of values for the variables that satisfy all equations. Geometrically, this means the lines (or hyperplanes) overlap or intersect along a line or plane.

Methods for Solving Systems of Linear Equations

Several methods exist for solving systems of linear equations. Here are some of the most common and widely used techniques:

1. Substitution Method: This method involves solving one equation for one variable and then substituting that expression into the other equation(s). This reduces the number of variables and equations until you can solve for one variable. You then back-substitute to find the values of the other variables.

2. Elimination Method (also known as the Addition Method): This method involves multiplying one or more equations by constants so that the coefficients of one of the variables are opposites. Then, you add the equations together, which eliminates that variable. This leaves you with a system with fewer variables, which you can solve. You then back-substitute to find the values of the other variables.

3. Gaussian Elimination: This is a systematic method for solving systems of linear equations by transforming the augmented matrix (a matrix formed by combining the coefficient matrix and the constant vector) into row echelon form or reduced row echelon form using elementary row operations.

4. Gauss-Jordan Elimination: This is an extension of Gaussian elimination that transforms the augmented matrix into reduced row echelon form. In reduced row echelon form, the leading coefficient (pivot) of each row is 1, and all other entries in the column containing a pivot are 0. This method directly gives the solution to the system.

5. Matrix Inversion: If the coefficient matrix A is invertible (i.e., has an inverse matrix A⁻¹), then the solution to the system Ax = b is given by x = A⁻¹b. This method is efficient when you need to solve multiple systems with the same coefficient matrix but different constant vectors.

6. Cramer's Rule: This method uses determinants to find the solution to a system of linear equations. It involves calculating several determinants derived from the coefficient matrix. Cramer's rule is generally less efficient than Gaussian elimination for large systems but can be useful for small systems or when you only need to find the value of a single variable.

7. Iterative Methods (e.g., Jacobi, Gauss-Seidel): These methods start with an initial guess for the solution and then iteratively refine the solution until it converges to the true solution. Iterative methods are particularly useful for solving large, sparse systems (systems where most of the coefficients are zero).

Examples of Solving Systems of Linear Equations

Let's illustrate these methods with several examples:

Example 1: Substitution Method

Solve the following system of equations:

• x + y = 5
• 2x - y = 1

Solution:

1. Solve the first equation for x: x = 5 - y
2. Substitute this expression for x into the second equation: 2(5 - y) - y = 1
3. Simplify and solve for y: 10 - 2y - y = 1 => -3y = -9 => y = 3
4. Substitute y = 3 back into the equation x = 5 - y: x = 5 - 3 => x = 2

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

Example 2: Elimination Method

Solve the following system of equations:

• 3x + 2y = 8
• x - 2y = 0

Solution:

1. Notice that the coefficients of y are already opposites (2 and -2).
2. Add the two equations together: (3x + 2y) + (x - 2y) = 8 + 0 => 4x = 8
3. Solve for x: x = 8 / 4 => x = 2
4. Substitute x = 2 back into the second equation: 2 - 2y = 0 => -2y = -2 => y = 1

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

Example 3: Gaussian Elimination

Solve the following system of equations using Gaussian elimination:

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

Solution:

1. Write the augmented matrix:

   [ 1  1  1 | 6 ]
   [ 2 -1  1 | 3 ]
   [ 1  2 -1 | 2 ]
   

2. Perform row operations to get zeros below the leading 1 in the first column:

   • R2 = R2 - 2*R1
   • R3 = R3 - R1

   [ 1  1  1 | 6 ]
   [ 0 -3 -1 | -9 ]
   [ 0  1 -2 | -4 ]
   

3. Swap R2 and R3 to get a better pivot in the second column:

   [ 1  1  1 | 6 ]
   [ 0  1 -2 | -4 ]
   [ 0 -3 -1 | -9 ]
   

4. Perform row operation to get a zero below the leading 1 in the second column:

   • R3 = R3 + 3*R2

   [ 1  1  1 | 6 ]
   [ 0  1 -2 | -4 ]
   [ 0  0 -7 | -21 ]
   

5. Solve for z from the last row: -7z = -21 => z = 3

6. Substitute z = 3 into the second row: y - 2(3) = -4 => y - 6 = -4 => y = 2

7. Substitute y = 2 and z = 3 into the first row: x + 2 + 3 = 6 => x = 1

Therefore, the solution is x = 1, y = 2, and z = 3.

Example 4: Gauss-Jordan Elimination

Solve the same system as in Example 3 using Gauss-Jordan elimination:

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

Solution:

1. Start with the augmented matrix (same as in Gaussian elimination):

   [ 1  1  1 | 6 ]
   [ 2 -1  1 | 3 ]
   [ 1  2 -1 | 2 ]
   

2. Perform the same row operations as in Gaussian elimination to get to row echelon form:

   [ 1  1  1 | 6 ]
   [ 0  1 -2 | -4 ]
   [ 0  0 -7 | -21 ]
   

3. Divide the last row by -7 to get a leading 1:

   [ 1  1  1 | 6 ]
   [ 0  1 -2 | -4 ]
   [ 0  0  1 | 3 ]
   

4. Now, perform row operations to get zeros above the leading 1s (pivots):

   • R2 = R2 + 2*R3
   • R1 = R1 - R3

   [ 1  1  0 | 3 ]
   [ 0  1  0 | 2 ]
   [ 0  0  1 | 3 ]
   

5. Finally, perform row operation to get a zero above the leading 1 in the second column:

   • R1 = R1 - R2

   [ 1  0  0 | 1 ]
   [ 0  1  0 | 2 ]
   [ 0  0  1 | 3 ]
   

The matrix is now in reduced row echelon form. The solution is directly read from the last column: x = 1, y = 2, and z = 3.

Example 5: Matrix Inversion

Solve the following system of equations using matrix inversion:

• 2x + y = 7
• x + 3y = 11

Solution:

1. Write the system in matrix form: Ax = b

   A = [ 2  1 ]
       [ 1  3 ]
   
   x = [ x ]
       [ y ]
   
   b = [ 7 ]
       [ 11 ]
   

2. Find the inverse of matrix A:

   • det(A) = (2 * 3) - (1 * 1) = 6 - 1 = 5
   • A⁻¹ = (1/det(A)) * [ 3 -1 ] [ -1 2 ]

   A⁻¹ = [ 3/5  -1/5 ]
          [ -1/5  2/5 ]
   

3. Calculate x = A⁻¹b:

   x = [ 3/5  -1/5 ] * [ 7 ]
       [ -1/5  2/5 ]   [ 11 ]
   
   x = [ (3/5)*7 + (-1/5)*11 ]
       [ (-1/5)*7 + (2/5)*11 ]
   
   x = [ 21/5 - 11/5 ]
       [ -7/5 + 22/5 ]
   
   x = [ 10/5 ]
       [ 15/5 ]
   
   x = [ 2 ]
       [ 3 ]
   

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

Example 6: Cramer's Rule

Solve the following system of equations using Cramer's Rule:

• x - 2y = 3
• 2x + y = 8

Solution:

1. Calculate the determinant of the coefficient matrix:

   D = | 1 -2 | = (1 * 1) - (-2 * 2) = 1 + 4 = 5
       | 2  1 |
   

2. Calculate the determinant Dx by replacing the first column of the coefficient matrix with the constant vector:

   Dx = | 3 -2 | = (3 * 1) - (-2 * 8) = 3 + 16 = 19
        | 8  1 |
   

3. Calculate the determinant Dy by replacing the second column of the coefficient matrix with the constant vector:

   Dy = | 1  3 | = (1 * 8) - (3 * 2) = 8 - 6 = 2
        | 2  8 |
   

4. Solve for x and y:

   • x = Dx / D = 19 / 5
   • y = Dy / D = 2 / 5

Therefore, the solution is x = 19/5 and y = 2/5.

Example 7: An Inconsistent System (No Solution)

Consider the system:

• x + y = 2
• x + y = 5

If you try to solve this using substitution or elimination, you'll quickly run into a contradiction. Subtracting the first equation from the second yields 0 = 3, which is false. This system has no solution because the two lines represented by these equations are parallel and never intersect.

Example 8: A Dependent System (Infinitely Many Solutions)

Consider the system:

• x + y = 3
• 2x + 2y = 6

Notice that the second equation is just a multiple of the first equation (multiplied by 2). This means the two equations represent the same line. There are infinitely many solutions. We can express the solution in terms of a parameter. Let y = t, where t is any real number. Then x = 3 - t. So, the solutions are of the form (3 - t, t). For example, (3, 0), (0, 3), (1, 2) are all solutions.

Choosing the Right Method

The best method for solving a system of linear equations depends on several factors, including the size of the system, the structure of the coefficient matrix, and the desired level of accuracy.

• For small systems (2x2 or 3x3) with simple coefficients, substitution, elimination, or Cramer's rule can be convenient.

• For larger systems, Gaussian elimination or Gauss-Jordan elimination are generally more efficient.

• If you need to solve multiple systems with the same coefficient matrix but different constant vectors, matrix inversion can be efficient.

• For large, sparse systems, iterative methods like Jacobi or Gauss-Seidel are often the best choice.

• If you only need to find the value of a single variable, Cramer's rule can be useful.

Applications of Systems of Linear Equations

Systems of linear equations are used extensively in various fields:

• Engineering: Solving for currents and voltages in electrical circuits, analyzing forces in structural mechanics, and modeling fluid flow.

• Computer Science: Solving linear systems is crucial in computer graphics (e.g., transformations, projections), machine learning (e.g., linear regression, support vector machines), and network analysis.

• Economics: Modeling supply and demand, analyzing market equilibrium, and performing input-output analysis.

• Data Analysis: Linear regression, a fundamental statistical technique, involves solving a system of linear equations to find the best-fit line or hyperplane for a set of data points.

• Operations Research: Linear programming, a technique for optimizing a linear objective function subject to linear constraints, relies on solving systems of linear equations and inequalities.

• Cryptography: Certain encryption algorithms rely on solving systems of linear equations in modular arithmetic.

Conclusion

Understanding and mastering the techniques for solving systems of linear equations is a cornerstone of mathematical and scientific literacy. From substitution and elimination to Gaussian elimination and matrix inversion, each method offers a unique approach to tackling these problems. By carefully considering the characteristics of the system, you can choose the most efficient and effective method to find the solution. The wide-ranging applications of linear systems in engineering, computer science, economics, and other fields underscore the importance of this fundamental concept. Proficiency in solving these systems empowers you to model and analyze real-world phenomena, making informed decisions and solving complex problems.