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System of Two Linear Equations in Two Unknowns

A system of two linear equations in two unknowns is a system of two equations of the form
a1 x b1y = c1
a2 x b2y = c2
where a1, a2, b1, b2, c1, c2 are arbitrary real numbers.

The solution set of the system of linear equation in two unknowns is a pair of real numbers (x0, y0) which satisfies each of the equations of the system. In general,
Description: 1210.png
x is defined if a1b2 − a2b1 ≠ 0 (i.e., for the existence of a unique solution of the system of linear equations)
 
In other words, the system of linear equations has a unique solution if (a1b2 − a2b1) ≠ 0
Description: 1217.png
 
Now let us consider the geometric interpretation of the discussing given above. We know that the set of points in a plane whose coordinates satisfy an equation of the form ax by = c, where either a or b are non-zero, constitutes a straight line. And to solve a system in which every equation has at least one unknown means to find the common point of two straight lines. Therefore, a system of linear equations has a unique solution if the lines intersect.
Example

Consider the system of linear equations.
4x + 6y = 10 and 4x − 2y = 2

 


Point (1, 1) is the intersection of two straight lines. Hence, x = 1, 
y = 1 is the unique solution of system of linear equations.
 
We can say that the system of linear equations has a unique solution if a1b2 − a2b1 ≠ 0.

What if a1b2 − a2b1 = 0?
If a1b2 − a2b1 = 0
or a1b2 = a2b1
or Description: 1224.png, then there are two situations:
 
Situation (i)
Description: 1231.png
If situation (i) exists, then the system of linear equations has an infinite number of solutions.
 
Situation (ii)
Description: 1238.png
If condition (ii) exists, then the system of linear equations has no solutions.
 
For example, consider following equations:
x + y = 10 ....(1)
2x + 2y = 25 ....(2)
 
In the given equations, a1/a2 = b1/b2, hence, unique solution is not possible. To have a better understanding, it can be seen that LHS of equation (1) × 2 will give us the LHS of equation (2). Hence, there will not be any point of intersection of these two graphs drawn on XY axis. In other words lines will be parallel.

Summarizing the whole discussion

In geometric terms In algebraic terms
(i) The lines intersect System of linear equations has a unique solution, i.e., Description: 1245.png (known as system is determinate)
(ii) The lines are The system is inconsistent, i.e., parallel the system of linear equa-tions has no solution, i.e.,
(iii) The lines are System of linear equations has coincident infinitely many solutions, i.e., Description: 1252.png (known as system is indeterminate)




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