System of Linear Equations

A system of linear equations is a set of two or more linear equations involving the same variables. Each equation represents a straight line or a plane and the solution to the system is the set of values for the variables that satisfy all equations simultaneously.

Here is simple example of system of linear equations:

General Form of System of Linear Equations

A system of linear equations consists of multiple linear equations involving the same set of variables. It can be represented as follows:
\begin{aligned}a_{11}x_1 + a_{12}x_2 + a_{13}x_3 + \dots + a_{1n}x_n &= b_1 \\a_{21}x_1 + a_{22}x_2 + a_{23}x_3 + \dots + a_{2n}x_n &= b_2 \\&\vdots \\a_{n1}x_1 + a_{n2}x_2 + a_{n3}x_3 + \dots + a_{nn}x_n &= b_n\end{aligned}

This represents a system of n linear equations in n variables x₁, x₂, x₃,…., x_n.

Where,

• a₁₁, a₁₂, …, a₂₁, a₂₂,…, a_n1, a_n2,…, a_nn are the coefficients of variables, x₁, x₂,…., x_n.
• b₁ + b₂ + b₃ + …. + b_n are the constants on the right-hand side of each equation.

Matrix Equation

These equations can be written in matrix form as AX = B, where:

• A is the coefficient matrix,
• X is the column vector of variables \left[ x_1, x_2, x_3, \dots, x_n \right]^T ,
• B is the column vector of constants \left[ b_1 + b_2 + b_3 + \dots + b_n \right]^T .

A =\begin{bmatrix} a_{11} & a_{12} & a_{13}& \cdots & a_{1n} \\ a_{21} & a_{22} & a_{23}& \cdots & a_{2n} \\ a_{31} & a_{32} & a_{33}&\cdots & a_{3n}\\\vdots & \vdots & \vdots & \vdots & \vdots \\ a_{m1} & a_{m2} & a_{m3} &\cdots & a_{mn} \end{bmatrix} , X = \begin{bmatrix} x_1 \\ x_2 \\ x_3 \\\vdots \\x_{n}\end{bmatrix} and\ B = \begin{bmatrix} b_1 \\ b_2 \\ b_3 \\\vdots \\b_{m}\end{bmatrix}

Solving the system involves finding the values of x₁, x₂, x₃,…., x_n that satisfy all equations simultaneously.

Types of system of linear equations

1. Homogeneous System (AX=0)
2. Non-Homogeneous System (AX=B)

The solution of these systems depends on the rank of the coefficient matrix A and the rank of the augmented matrix [A :B].

1) System of Homogeneous linear equations AX = 0

1. X = 0 is always a solution: means all the unknowns has same value as zero. (This is also called trivial solution)
2. If P(A) = number of unknowns : Unique solution ( P(A) = Rank of matrix A).
3. If P(A) < number of unknowns : Infinite number of solutions.

Since a homogeneous system always has at least one solution (X = 0), it is always consistent.

Example of Homogeneous System in three variable:

x + y - z = 0
x + y + z = 0
x - y + 2z = 0

2) System of Non-Homogeneous linear equations AX = B

1. If P[A:B] ≠ P(A), No solution.
2. If P[A:B] = P(A) = Number of unknown variables, unique solution.
3. If P[A:B] = P(A) ≠ Number of unknown, infinite number of solutions.

Example of Non - Homogeneous System in three variable

x + y - 2z = 6
x - 6y + z = 9
2x - y + 2z = 2

Geometric interpretation

For a system of two linear equations with two variables (x and y), solving the system involves finding the point of intersection of the two lines represented by the equations on the xy-plane. There are three possibilities:

1) The lines intersect at one point: This point is the unique solution to the system.

2) The lines are parallel: They never intersect, so there is no solution. The system is called inconsistent.

3) The lines are coincident: They are the same line, so every point on the line is a solution. There are infinitely many solutions.

The solution set to a system of two linear equations in two variables can be a single point, the empty set (no solution) or an infinite set of points (a line).

How to Solve System of Linear Equations?

The subsequent techniques for solving the system of linear equations AX = B are viable solely under the condition that the coefficient matrix A is non-singular, meaning |A| ≠ 0 (determinant is not zero).

These methods include:

• Cramer's Rule: This is a mathematical theorem used to solve systems of linear equations using determinants, providing a straightforward formula for each variable in the system.
• Inverse Method: This involves finding the inverse of the coefficient matrix in a system of linear equations and multiplying it by the constant vector to solve for the variables.
• Gauss Elimination Method: This systematically eliminates variables from a system of linear equations using row operations to transform the system into an upper triangular matrix for easier back substitution.
• LU Decomposition Method of Factorization: This a matrix into the product of a lower triangular matrix (L) and an upper triangular matrix (U), simplifying the solution of linear systems by solving two simpler triangular systems.

Applications System of Linear Equations in Engineering

Systems of linear equations are widely used in various engineering disciplines:

• Structural Analysis: In civil and mechanical engineering, they are used to analyze forces in structures, determine displacements and design stable frameworks.
• Electrical Circuit Analysis: In electrical engineering, Kirchhoff's laws lead to systems of linear equations that are used to analyze currents and voltages in electrical circuits.
• Control Systems: In control engineering, linear equations model dynamic systems and are used to design controllers that ensure desired system behavior.
• Optimization Problems: In industrial engineering and operations research, systems of linear equations arise in linear programming problems used to optimize production, transportation and resource allocation.