Question

Check whether the following equations are consistent or inconsistent, solve them graphically.
\[x + y = 5\]
\[2x + 2y = 10\]

Answer 1

Hint: For checking whether the pair of linear equations are consistent or inconsistent, we try to obtain values of x and y. Here, we are given two equations and we will check using the conditions for consistency. If the pair of linear equations is consistent, then the lines either intersect or coincide. In that, the lines are coincident and have an infinite set of solutions. Also, we will draw the graph and thus, we will get the final output.

Complete step-by-step answer:
We are given two equations,
\[x + y = 5\] and \[2x + 2y = 10\]
We will write the equations in the general form as:
\[x + y - 5 = 0\] ----- (1)
\[2x + 2y - 10 = 0\] ----- (2)
Now, we check if the given equations are consistent or not by using the coefficient analysis method for determination of a consistent system. In this method the coefficients of x and y i.e. ‘a’ and ‘b’, are compared.
Thus, the condition for consistency is
\[\therefore \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}}\]
Here, for the equation (1) \[{a_1} = 1\] , \[{b_1} = 1\] and \[{c_1} = - 5\]
And, for the equation (2) \[{a_2} = 2\] , \[{b_2} = 2\] and \[{c_2} = - 10\]
Substituting these values, we will get,
\[ \Rightarrow \dfrac{1}{2} = \dfrac{1}{2} = \dfrac{{ - 5}}{{ - 10}}\]
\[ \Rightarrow \dfrac{1}{2} = \dfrac{1}{2} = \dfrac{1}{2}\]
Thus, the lines are coincident and the pair of equations is dependent and consistent.
The graph for the given two equations, is as below:

Hence, the given two equations are consistent and dependent because the lines have infinitely many solutions and when you graph the equations, both equations represent the same line.

Note: A system of two linear equations can have one solution, an infinite number of solutions, or no solution. Systems of equations can be classified by the number of solutions. If a system has at least one solution, it is said to be consistent. If a consistent system has exactly one solution, it is independent. If a consistent system has an infinite number of solutions, it is dependent. If a system has no solution, it is said to be inconsistent. The graphs of the lines do not intersect, so the graphs are parallel and there is no solution.