check the consistency of the system of equations x+2y+5=0 and 3x - 6y +1=0
Answers
Answer :
Consistent
Note:
★ A linear equation in 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 pair of linear equations are :
x + 2y + 5 = 0
3x - 6y + 1 = 0
Now ,
Comparing the given equations with the general equations ax + by + c = 0 and a'x + b'y + c' = 0 respectively , we have ;
a = 1
a' = 3
b = 2
b' = -6
c = 5
c' = 1
Thus ,
a/a' = 1/3
b/b' = 2/-6 = -1/3
c/c' = 3/1 = 3
Clearly ,
a/a' ≠ b/b'
Thus ,
The given lines are intersecting .
Hence ,
The given system of linear equations is consistent .
Answer:
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