2x-(a-4)y=2b+1, 4x-(a-1)y=5b-1 find a and b infinite solutions
Answers
Answer :
a = 7 , b = 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 :
Here ,
The given equations are ;
2x - (a - 4)y = 2b + 1
4x - (a - 1)y = 5b - 1
The given equations can be rewritten as ;
2x - (a - 4)y - (2b + 1) = 0 -------(1)
4x - (a - 1)y - (5b - 1) = 0 --------(2)
From eq-(1) , we have ;
A = 2
B = -(a - 4)
C = -(2b + 1)
From eq-(2) , we have ;
A' = 4
B' = -(a - 1)
C' = -(5b - 1)
Now ,
• A/A' = 2/4 = 1/2
• B/B' = -(a - 4)/-(a - 1) = (a - 4)/(a - 1)
• C/C' = -(2b + 1)/-(5b - 1) = (2b + 1)/(5b - 1)
For the given equations to have infinitely many solutions , A/A' = B/B' = C/C' .
Thus ,
1/2 = (a - 4)/(a - 1) = (2b + 1)/(5b - 1)
• Considering 1/2 = (a - 4)/(a - 1)
=> 1/2 = (a - 4)/(a - 1)
=> a - 1 = 2(a - 4)
=> a - 1 = 2a - 8
=> 2a - a = 8 - 1
=> a = 7
• Considering 1/2 = (2b + 1)/(5b - 1)
=> 1/2 = (2b + 1)/(5b - 1)
=> 5b - 1 = 2(2b + 1)
=> 5b - 1 = 4b + 2
=> 5b - 4b = 2 + 1
=> b = 3