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Systems of Two Linear Equations in Two Unknowns. Geometric Interpretation
Consider the systems of two linear equations in two unknowns.
a1 x + b1 y = c1
Suppose that none of a1 , a2 , b1 , b2 , c1 , c2 is zero.

a2 x + b2 y = c2
a1 x + b1 y = c1 represents the standard form of the equation of a line (L1).
a
c
a
In the slope intercept form: y = − 1 x + 1 if b1 ≠ 0 the slope is m1 = − 1
b1
b1
b1
a2 x + b2 y = c2 represents the standard form of the equation of a line (L2)
a
c
a
or in the slope intercept form y = − 2 x + 2 if b2 ≠ 0 with the slope m2 = − 2 .
b2
b2
b2
Note: If at least one of c1 or c2 is nonzero then the system
a1 x + b1 y = c1
in called nonhomogeneous.

a2 x + b2 y = c2
CASE 1: The system has unique solution
If m2 ≠ m1 the 2 lines intersect in one point P (X’, Y’)
a
b
a
b
But m2 ≠ m1 means 1 ≠ 1 or 1 ≠ 1 or the quantity ∆ = a1b2 − a2b1 ≠ 0
a2 b2
a2 b2
In this case the system has a unique solution (is consistent), and the equations are
independent (one can NOT obtain one equation dividing the other by some constant).
y
P
Y’
Line 1
X’
x
Line 2
The unique solution is represented by the coordinates X’, Y’ of the point of intersection
of the lines L1 and L2.
CASE 2 The system has no solution
a
b
a
b
c
If m2 = m1 i.e. 1 = 1 (or the quantity ∆ = a1b2 − a2b1 = 0 ) but 1 = 1 ≠ 1 the 2 lines
a2 b2
a2 b2 c2
are parallel and therefore the system has no solution (is inconsistent).The equations are
independent.
y
x
CASE 3 The system has infinitely many solutions
a1 b1 c1
= =
the two lines are identical (overlap) and therefore the system has infinitely
a2 b2 c2
many solution (is consistent).
The solution is the set of all points on the line a1 x + b1 y = c1 .There are infinitely many
such points on this, and their( x, y);their x and y coordinates represent the solution of the
system.
a
c
Because a1 x + b1 y = c1 means y = − 1 x + 1 the solution can be written as follows:
b1
b1
a
c
(x = any real value, y = − 1 x + 1 ) The system is consistent and the equations are
b1
b1
dependent. Both equations represent the same line
y