Question

The given pair of linear equations in two variables is consistent or inconsistent? If it is consistent, obtain the solution graphically.
\[2x + y - 6 = 0\]
 $ 4x + 2y - 4 = 0 $

Answer 1

Hint: In the given question, we are provided with a pair of equations in two variables and we have to find whether the equations are consistent or not. We can find this by comparing the coefficients and constant terms of the two equations. Consistent system of linear equations in two variables is the system which has one or more than one solution. Inconsistent system, on the other hand, has no solution for the two linear equations in two variables.

Complete step by step solution:
So, the given system of simultaneous linear equations in two variables is:
\[2x + y - 6 = 0 - - - - \left( 1 \right)\]
$ 4x + 2y - 4 = 0 - - - - \left( 2 \right) $
Now, we know that the general form of a linear equation in two variables is $ ax + by + c = 0 $ .
Now, for the equation $ \left( 1 \right) $ , we have,
 $ {a_1} = 2 $ , $ {b_1} = 1 $ and $ {c_1} = - 6 $ .
Also, for the equation $ \left( 2 \right) $ , we have,
 $ {a_2} = 4 $ , $ {b_2} = 2 $ and $ {c_2} = - 4 $ .
Now, we know that there are three types of systems of linear equations in two variables. One is the system where there is no solution for the linear equations in two variables. In this case, both the equations represent parallel lines on a Cartesian plane. Such a system is known as an inconsistent system of linear equations in two variables. For system to have no solution at all, the coefficient and constants of equations should be in ratio like $ \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}} $ .
Second is the system of linear equations where there are infinitely many solutions of the equations. In this case, both equations represent the same or overlapping lines on a Cartesian plane. Such a system of equations is a consistent system of linear equations as it possesses more than one solution for the linear equations. For system to have infinitely many solution, the coefficient and constants of equations should be in ratio like $ \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}} $ .

Thirdly, the system of linear equations in two variables where there is only one and unique solution for the equation. In this case, both equations represent two intersecting lines on the Cartesian plane. Such a system is also a consistent system of linear equations in two variables. For system to have one unique solution, the coefficient and constants of equations should be in ratio like $ \dfrac{{{a_1}}}{{{a_2}}} \ne \dfrac{{{b_1}}}{{{b_2}}} $ .
Hence, to find if the given system of equations is consistent or inconsistent, we compare the coefficients and constants of the given equations.
Now, $ \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{2}{4} = \dfrac{1}{2} $
 $ \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{1}{2} $
 $ \dfrac{{{c_1}}}{{{c_2}}} = \dfrac{{ - 6}}{{ - 4}} = \dfrac{3}{2} $
Now, we can see that $ \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}} $ .
Now, we know that if the ratio of the coefficients and constants of two equations is of the form $ \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} \ne \dfrac{{{c_1}}}{{{c_2}}} $ , then the linear equations in two variables have no roots.
Hence, the system of linear equations in two variables is inconsistent.

Note: We must have a grip over the topic of linear equations in two variables in order to solve such a question within a limited time frame. We must ensure accuracy in arithmetic and calculations to be sure of our answer. General form of a linear equation in two variables must be remembered. One should always convert the equations into the general form before comparing the coefficients to decide the nature of the system of linear equations.