System of Linear Equations
Last Updated :
05 Feb, 2025
In mathematics, a system of linear equations consists of two or more linear equations that share the same variables. These systems often arise in real-world applications, such as engineering, physics, economics, and more, where relationships between variables need to be analyzed. Understanding how to solve these systems helps in finding the intersection points of lines or planes, which represent the solution to the equations.
- A system of linear equations consists of multiple linear equations with shared variables, where each equation represents a line, plane, or higher-dimensional surface based on the number of variables.
- The solution is the set of variable values that satisfy all equations, often corresponding to the points of intersection of the represented lines or planes.
Here is simple example of system of linear equations:
General form
A system of linear equations consists of multiple linear equations involving the same set of variables. It can be represented as follows:
a11x1 + a12x2 + a13x3 + …. + a1nxn = b1
a21x1 + a22x2 + a23x3 + …. + a2nxn = b2
…………………………………..
…………………………………..
an1x1 + an2x2 + an3x3 + …. + annxn = bn
This represents a system of n linear equations in n variables x1, x2, x3,…., xn.
Where,
- a11, a12, …, a21, a22,…, an1, an2,…, ann are the coefficients of variables, x1, x2,…., xn.
- b1 + b2 + b3 + …. + bn 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 [x1, x2, x3,…., xn ]T ,
- B is the column vector of constants [b1 + b2 + b3 + …. + bn ]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 & \ddots & \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,…., xn that satisfy all equations simultaneously.
Solution of a System of Linear Equation
A system of linear equations can have three possible types of solutions:
- No Solution (Inconsistent System)
- Unique Solution (Consistent and Determinate System)
- Infinite Solutions (Consistent and Indeterminate System)
The existence and nature of solutions depend on the properties of the coefficient matrix A and the augmented matrix [A:B].
A system of linear equations can be categorized into two types:
- Homogeneous System (AX=0)
- 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
- X = 0 is always a solution; means all the unknowns has same value as zero. (This is also called trivial solution)
- If P(A) = number of unknowns : Unique solution ( P(A) = Rank of matrix A).
- 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
- If P[A:B] ≠ P(A), No solution.
- If P[A:B] = P(A) = Number of unknown variables, unique solution.
- 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): Since each linear equation represents a line on the xy-plane, solving a system of two equations means finding the points where the lines intersect. 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).
For n variables, each linear equation determines a hyperplane in n-dimensional space. The solution set is the intersection of these hyperplanes, and is a flat, which may have any dimension lower than n.
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
Cramer’s rule is applicable only when certain conditions are satisfied. The important condition of Cramer’s rules are,
System of Linear EquationsApplications System of Linear Equations in Engineering
Systems of linear equations are widely used in various engineering disciplines:
- Structural Analysis: In civil and mechanical engineering, systems of linear equations 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.
Examples on System of Linear Equations
Example 1: Solve the system:
Solution:
From equation first : y = 5 - x
Substitute into equation second : 2x - (5 - x) = 1
3x - 5 = 1
3x = 6
x = 2
Substitute back to find y: y = 5 - 2 = 3
Answer: (x,y)= (2, 3)
Example 2: Solve using substitution:
Solution:
From equation second : x = 1 + y
Substitute into equation first : 3(1 + y) + 2y = 13
3 + 3y + 2y = 13
5y = 10
y = 2
Substitute back: x = 1 + 2 = 3
Answer: (x,y)=(3, 2)
Example 3: Solve using elimination:
Solution:
Add the equations: 6x = 10
x = 5/3
Substitute into first equation: 2(5/3) + 3y = 8
10/3 + 3y = 8
3y = 14/3
y = 14/9
Answer: (5/3, 14/9)
Example 4: Determine if the system is consistent:
Solution:
Multiply the first equation by 2: 4x + 2y = 8
This is identical to the second equation, so the system is consistent with infinitely many solutions.
Example 5: Solve graphically:
Solution:
Plot both lines. They intersect at (1, 3).
Answer: (1, 3)
Example 6: Solve using Cramer's Rule:
Solution:
D = \begin{vmatrix}3&2\\1&1\\\end{vmatrix} = 3(1) - 2(1) = 1
Dx = \begin{vmatrix}14&2\\6&1\\\end{vmatrix} = 14(1) - 2(6) = 2
Dy = \begin{vmatrix}3&14\\1&6\\\end{vmatrix}= 3(6) - 14(1) = 4
x = Dx/D = 2/1 = 2
y = Dy/D = 4/1 = 4
Answer: (2, 4)
Practice Problems on System of Linear Equations
Problem 1. Solve the system:
Problem 2. Determine if the system is consistent:
Problem 3. Solve using substitution:
Problem 4. Solve the 3x3 system:
- x + y - z = 2
- 2x + y + z = 7
- x - y + 2z = 4
Problem 5. Solve graphically:
Problem 6. Use Cramer's Rule to solve:
Problem 7. Determine the nature of the system:
Problem 8. Solve using Gauss Elimination:
- 2x+3y+z=10
- 4x−y+5z=5
- x+2y−3z=−4
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