To verify the condition consitency / inconsistency for a pair of linear equation in two variable by graphical method
Dear Student,
A pair of linear equations in two variables in general is:
a1x+b1y+c1 = 0 ......(1)
a2x+b2y+c2 = 0........(2)
We can find the solution to these equations by graphical method.
First sketch the graph of pair of linear equations in two variables, draw two lines representing the equations by taking random values for x and y.Then the following cases are possible:
i) If both the lines intersect at a point, then there exists a unique solution to the pair of linear equations. In such a case, the pair of linear equations is said to be consistent.
Algebraically, if a1a2 ≠ b1b2 then, the linear equations’ pair is consistent.
ii) Consider two lines having equation to be-
a1x+b1y+c1 = 0 and
a2x+b2y+c2 = 0
Let these lines coincide with each other, then there exist infinitely many solutions since a line consists of infinite points. In such a case, the pair of linear equations is said to be dependent and consistent. As represented in the graph below, the pair of lines are coincident and therefore, dependent and consistent.
Algebraically, when a1a2 = b1b2 = c1c2 , then the lines are coincident and the pair of equations is dependent and consistent.
iii) Consider the equation of the lines to be-
a1x+b1y+c1 = 0 and
a2x+b2y+c2 = 0
Let both the lines to be parallel to each other, then there exists no solution, because the lines never intersect.
Algebraically, for such a case, a1a2 = b1b2 ≠ c1c2 and the pair of linear equations in two variables is said to be inconsistent.
As shown in the graph above, the pair of lines a1x+b1y+c1 = 0 and a2x+b2y+c2 = 0 are parallel to each other. Therefore, there exists no solution for such a pair.
Regards
Manshi Arora
Dear Student,
A pair of linear equations in two variables in general is:
a1x+b1y+c1 = 0 ......(1)
a2x+b2y+c2 = 0........(2)
We can find the solution to these equations by graphical method.
First sketch the graph of pair of linear equations in two variables, draw two lines representing the equations by taking random values for x and y.Then the following cases are possible:
i) If both the lines intersect at a point, then there exists a unique solution to the pair of linear equations. In such a case, the pair of linear equations is said to be consistent.
Algebraically, if a1a2 ≠ b1b2 then, the linear equations’ pair is consistent.
ii) Consider two lines having equation to be-
a1x+b1y+c1 = 0 and
a2x+b2y+c2 = 0
Let these lines coincide with each other, then there exist infinitely many solutions since a line consists of infinite points. In such a case, the pair of linear equations is said to be dependent and consistent. As represented in the graph below, the pair of lines are coincident and therefore, dependent and consistent.
Algebraically, when a1a2 = b1b2 = c1c2 , then the lines are coincident and the pair of equations is dependent and consistent.
iii) Consider the equation of the lines to be-
a1x+b1y+c1 = 0 and
a2x+b2y+c2 = 0
Let both the lines to be parallel to each other, then there exists no solution, because the lines never intersect.
Algebraically, for such a case, a1a2 = b1b2 ≠ c1c2 and the pair of linear equations in two variables is said to be inconsistent.
As shown in the graph above, the pair of lines a1x+b1y+c1 = 0 and a2x+b2y+c2 = 0 are parallel to each other. Therefore, there exists no solution for such a pair.
Regards
Manshi Arora
