System of Linear Equations

System of Linear Equations

We have already discussed the linear equations under the topic Quadratic Equations. The set of n (> 2) linear equations is called the system of linear equations and this system is said to be consistent if it has at least one solution.

Illustration:

Check the consistency of the following system of equation.

(i)     x + 2x = 4

        2x + 4y = 9

This is inconsistent because it has no solution i.e. there is no value of x and y which satisfies both the equations.

(ii)    x + y = 4

        X - y = 0

This is consistent because it has a solution x = 2 and y = 2

Solving System of Linear Equations by using Determinants

There are several methods to solve the system of linear equations but determinant is one of the best mathematical tool from which we can solve the system of linear equations very easily.

CRAMER'S RULE

Case I:      System of linear equations in two variables.

                Let, us have the system of equations

                a₁x + b₁ y + c₁ = 0

                 a₂x + b₂ y + c₂ = 0, where a₁/a₂ ≠ b₁/b₂

                 Solving by cross multiplication, we get,

Case II:    System of linear equations in three variables:

                Let, us have the system of equations.

                a₁ x + b₁ y + c₁ = d₁

                a₂ x + b₂ y + c₂ = d₂

                a₃ x + b₃ y + c₃ = d₃

                .·.    x = Δ₁/Δ

                Similarly

                y = Δ₂/Δ and z = Δ₃/Δ, where Δ ≠ 0.