Linear Equations

Consider these two equations as we explore the methods of solving simultaneous linear equations :

Linear equations of 2 variables: 

 
Say we have two equations : ax + by = c, and, px + qy = r,
where a, b, c, p, q and r are constants, and, x and y are two variables.

 

Now consider the following three cases: -

If a/p = b/q = c/r, then the equations will give infinite number of solutions (graphically, the equations denote the same straight line).

If a/p = b/q ≠ c/r, then we have no solution (graphically, the equations denote two parallel straight lines that do not intersect).

If a/p ≠ b/q, then we have one unique solution (graphically the equations denote two straight lines intersecting at a particular point).

A system of linear equations in two variables can be solved by the following methods:-

        

Consider these two equations as we explore the methods of solving simultaneous linear equations :

 

Substitution method :

Cross multiplication or Cramer’s rule:

Comparison method:

 

Elimination Method: