Section 2

The existence of a unique solution

A set of linear simultaneous equations may have (i) a unique solution, (ii) no solution, or (iii) infinitely many solutions.

In a set of linear simultaneous equations, a unique solution exists if and only if, (a) the number of unknowns and the number of equations are equal, (b) all equations are consistent, and (c) there is no linear dependence between any two or more equations, that is, all equations are independent.

In a system of linear simultaneous equations if one or more equations are inconsistent, the system does not have any solution. For example, if in a set of linear simultaneous equations with two equations and two unknowns, one equation is \(x+y=2\) and another equation is \(3x+3y=5\), these two equations are inconsistent within the given system. They are inconsistent because if \(x+y=2\), then \(3x+3y\) must be \(6\), not \(5\). One cannot solve the system of linear simultaneous equations \(x+y=2\) and \(3x+3y=5\) as they are inconsistent.

Graphically, the solution of two linear simultaneous equations in two unknowns is equivalent to finding where the lines of the two equations cross. If these two equations are inconsistent, corresponding lines in the Cartesian plane are parallel and will never cross (See Practice Question 2).

In a system of linear simultaneous equations if all equations are consistent, but (a) the number of independent equations is less than the number of unknowns, and/or (b) there exists a linear dependence between two or more equations in the system, there may exists infinitely many solutions that satisfy the system.

Linear dependence, for example between two linear equations, refers to a situation when one equation in the system is a multiple of another equation. For example, equations \(y=x+2\) and \(2y=2x+4\) are linearly dependent as the later can be obtained by multiplying the former equation by \(2\).

Consider a more general example. Suppose two linear simultaneous equations are: $$a_{11}x_{1} + a_{12}x_{2} = b_{1}$$ $$a_{21}x_{1} + a_{22}x_{2} = b_{2}$$ Where, \(a_{ij}\) is the coefficient, \(x_j\) is the variable, and \(b_i\) is the constant. In this system if \(a_{1j}=ka_{2j}\) and \(b_1=kb_2\), where \(k\) is a constant, the equations are linearly dependent. Graphically the lines representing the graphs of two equations coincide if the equations are linearly dependent, and every point on either line is a solution. [ See Practice Question 3]

One interesting form of linear dependence may arise in a system of \(m×n\) linear simultaneous equations when one equation is the sum or difference of more than one equation in the system. For example, equations (i) \(x+y+z=10\),(ii) \(2x-2y-2z=4\), and (iii) \(3x-y-z=14\) have linear dependence. (Why?)

To sum up, consider a system of linear simultaneous equations where all equations are consistent, however, due to the linear dependence between some equations the number of independent equations is less than the number of unknowns. Such a system has infinitely many solutions.

It follows from the discussion in this section is that two linear simultaneous equations in two unknowns can have a unique solution, no solution or infinitely many solutions and this is true for every system of linear simultaneous equations with \(m\) equations and \(n\) unknowns. To read more about the existence of a unique solution, inconsistency, and linear dependence, please see the recommended books.