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Qᴜᴇsᴛɪᴏɴ :-
Examine weather the following system of Equations are consistent or inconsistent .
1) x - 3y = 3 , 3x - 9y = 2
2) ax - 2by = a , bx + 2ay = b , when a ≠ 0, b ≠ 0 .
3) 2ax + by = a , 4ax + 2by = 2a , when a ≠ 0, b ≠ 0 .
ᴄᴏɴᴄᴇᴘᴛ ᴜsᴇᴅ :-
• A linear equation in two variables represents a straight line in 2D Cartesian plane .
• If we consider two linear equations in two variables, say :-
➻ a1x + b1y + c1 = 0
➻ a2x + b2y + c2 = 0
Then :-
✪ Both the straight lines will coincide if :-
a1/a2 = b1/b2 = c1/c2
➻ In this case , the system will have infinitely many solutions.
➻ If a consistent system has an infinite number of solutions, it is dependent and consistent.
✪ Both the straight lines will be parallel if :-
a1/a2 = b1/b2 ≠ c1/c2.
➻ In this case , the system will have no solution.
➻ If a system has no solution, it is said to be inconsistent.
✪ Both the straight lines will intersect if :-
a1/a2 ≠ b1/b2.
➻ In this case , the system will have an unique solution.
➻ If a system has at least one solution, it is said to be consistent..
____________________
Sᴏʟᴜᴛɪᴏɴ :-
1) x - 3y = 3 , 3x - 9y = 2
⟼ x - 3y - 3 = 0
⟼ 3x - 9y - 2 = 0
Comparing it with :-
⟼ a1x + b1y + c1 = 0
⟼ a2x + b2y + c2 = 0
we get :-
⟼ a1 = 1 , b1 = (-3) , c1 = (-3)
⟼ a2 = 3, b2 = (-9) , c2 = (-2)
Putting values Now we get :-
⟼ (a1/a2) = (b1/b2) = (c1/c2)
⟼ (1/3) = (-3)/(-9) = (-3)/(-2)
⟼ (1/3) = (1/3) ≠ (3/2)
Comparing Now we get :-
⟼ a1/a2 = b1/b2 ≠ c1/c2.
Hence, From Above Told Concept we can conclude that, Both lines are parallel, so, No solution & are inconsistent.
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2) ax - 2by = a , bx + 2ay = b , when a ≠ 0, b ≠ 0 .
⟿ ax - 2by - a = 0
⟿ bx + 2ay - b = 0
Comparing it with :-
⟿ a1x + b1y + c1 = 0
⟿ a2x + b2y + c2 = 0
we get :-
⟿ a1 = a , b1 = (-2b) , c1 = (-a)
⟿ a2 = b, b2 = (2a) , c2 = (-b)
Putting values Now we get :-
⟿ (a1/a2) = (b1/b2) = (c1/c2)
⟿ (a/b) = (-2b)/(2a) = (-a)/(-b)
⟿ (a/b) = (-b)/a = (a/b)
Comparing Now we get :-
⟿ (a/b) ≠ (-b)/a
Hence, From Above Told Concept we can conclude that, Both lines have a unique solution & are consistent.
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3) 2ax + by = a , 4ax + 2by = 2a , when a ≠ 0, b ≠ 0 .
➪ 2ax + by - a = 0
➪ 4ax + 2by - 2a = 0
Comparing it with :-
➪ a1x + b1y + c1 = 0
➪ a2x + b2y + c2 = 0
we get :-
➪ a1 = 2a , b1 = b , c1 = (-a)
➪ a2 = 4a, b2 = (2b) , c2 = (-2a)
Putting values Now we get :-
➪ (a1/a2) = (b1/b2) = (c1/c2)
➪ (2a/4a) = (b/2b) = (-a)/(-2a)
➪ (1/2) = (1/2) = (1/2)
Comparing Now we get :-
➪ a1/a2 = b1/b2 = c1/c2
Hence, From Above Told Concept we can conclude that, Both lines have infinitely many solution & are dependent consistent.