On compairing the ratios a1/a2 , b1/b2, c1/c2 find out whether the following consistent or inconsistent 12x+20y=8, 3x-5y=3
Answers
Answér :
Consistent
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 linear equations are ;
12x + 20y = 8
3x - 5y = 3
The given equations can be rewritten as ;
12x + 20y - 8 = 0
3x - 5y - 3 = 0
Clearly , we have ;
a1 = 12
a2 = 3
b1 = 20
b2 = -5
c1 = -8
c2 = -3
Now ,
a1/a2 = 12/3 = 4
b1/b2 = 20/-5 = -4
c1/c2 = -8/-3 = 8/3
Clearly ,
a1/a2 ≠ b1/b2
Thus ,
The given equations has an unique solution .