Let us determine the relations of ratios of the same variable and constants of the following pairs of equations and write whether the graphs of the equations will be parallel or intersecting or overlapping.
4x – 3y = 6
4y – 5x = –7
4x – 3y = 6 …(1)
4y – 5x = –7 …(2)
Let us express the equations (1) and (2) in the form of
ax + by +c = 0 where a and b can’t be 0 at the same time.
[In the first equation, we use a1, b1, c1 and in second equation, we use a2, b2, c2 ]
4x – 3y = 6 4y – 5x = –7
∴ 4x + (-3y) + (-6) = 0 ∴ –5x + 4y + 7= 0
Or 4 × x + (-3) × y + (-6) = 0 or, -5 × x + 4 × y + 7 = 0
Here a1 = 4, b1 = -3, c1 = -6 and a2 = -5, b2 = 4, c2 = 7
Comparing the ratio of 
, we get
, 
and 
Here 
. Therefore, it has one common solution. Graph of equations will intersect at a point.
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