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Answers
Step-by-step explanation:
gave any further questions
Answer :
a = -1/3 , b = -5/3
Note:
★ A linear equation is two variables represent a straight line .
★ The word consistent is used for the system of equations which consists any solution .
★ The word inconsistent is used for the system of equations which doesn't consists any solution .
★ Solution of a system of equations :
It refers to the possibile values of the variable which satisfy all the equations in the given system .
★ A pair of linear equations are said to be consistent if their graph ( Straight line ) either intersect or coincide each other .
★ A pair of linear equations are said to be inconsistent if their graph ( Straight line ) are parallel .
★ If we consider equations of two straight line Ax + By + C = 0 and
A'x + B'y + C' = 0 , then ;
• The lines are intersecting if
A/A' ≠ B/B' .
→ In this case , unique solution is found .
• The lines are coincident if
A/A' = B/B' = C/C' .
→ In this case , infinitely many solutions are found .
• The lines are parallel if
A/A' = B/B' ≠ C/C' .
→ In this case , no solution is found .
Solution :
The given linear equations are ;
2x - 3y = 7
(a - b)x + (a + b)y = 3a + b - 2
The given equations can be rewritten as ;
2x - 3y - 7 = 0
(a - b)x + (a + b)y - (3a + b - 2) = 0
Clearly ,
A = 2
A' = a - b
B = -3
B' = a + b
C = -7
C' = -(3a + b - 2)
Thus ,
For the given linear equations to have infinitely many solutions , we have ;
A/A' = B/B' = C/C'
Considering A/A' = B/B' , we get ;
=> 2/(a - b) = -3/(a + b)
=> 2×(a + b) = -3×(a - b)
=> 2a + 2b = -3a + 3b
=> 2a + 3a = 3b - 2b
=> 5a = b
=> b = 5a -------------(1)
Now ,
Considering B/B' = C/C'
=> -3/(a + b) = -7/-(3a + b - 2)
=> -3/(a + b) = 7/(3a + b - 2)
=> 3×(3a + b - 2) = 7×(a + b)
=> 9a + 3b - 6 = 7a + 7b
=> 9a - 7a + 3b - 7b = 6
=> 2a - 4b = 6
=> 2(a - 2b) = 6
=> a - 2b = 6/2
=> a - 2b = 3
=> a - 2×5a = 3 {using eq-(1)}
=> a - 10a = 3
=> -9a = 3
=> a = 3/-9
=> a = -1/3
Now ,
Putting a = -1/3 in eq-(1) , we get ;
=> b = 5a
=> b = 5 × (-1/3)
=> b = -5/3