Question

For what value of p, system of equations 2x + py = 8 and x + y = 6 have no solutions.

Answer 1

Hint: To solve the above question, we will first determine what kind of equations are given in the question. Then, we will find out what can be the different types of solutions when two linear equations are given. Then, we will make use of the fact that if one equation is \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\] and the other equation is \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0,\] then both the equation will have zero solution, only if it satisfies the following condition:
\[\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}\ne \dfrac{{{c}_{1}}}{{{c}_{2}}}\]
With the help of this, we will derive the relation between the coefficients of x and coefficients of y and constant term of both the equations. With the help of this relation, we will find the value of p.

Complete step-by-step answer:
Before solving the question, we must know that the given equations in the question are a pair of linear equations in two variables x and y. Now, there may be three types of solutions of a pair of linear equations in two variables: unique solution, no solution and infinite solutions. Now, we know that if two linear equations are of the form \[{{a}_{1}}x+{{b}_{1}}y+{{c}_{1}}=0\] and \[{{a}_{2}}x+{{b}_{2}}y+{{c}_{2}}=0,\] then we can say that if \[\dfrac{{{a}_{1}}}{{{a}_{2}}}=\dfrac{{{b}_{1}}}{{{b}_{2}}}\ne \dfrac{{{c}_{1}}}{{{c}_{2}}}\] then these two equations will have no solution. And we are given the system of equations, 2x + py = 8 and x + y = 6. Thus, we have the following equation.
\[\dfrac{2}{1}=\dfrac{p}{1}\ne \dfrac{-8}{-6}\]
Thus, we have,
\[\dfrac{2}{1}=\dfrac{p}{1}\]
\[\Rightarrow p=2\]
Thus, when the value of p will be 2, the equations will have no solution.

Note: The alternate way of solving the question is shown below. We will use the concept of parallel lines to solve this question according to which two parallel lines never intersect each other, i.e. they have no solution. Thus, if the given equations have no solution, then these lines should be parallel i.e. their slope should be the same. The slope of ax + by + c = 0 is \[\dfrac{-a}{b}.\] Thus,
\[\dfrac{-2}{p}=\dfrac{-1}{1}\]
\[\Rightarrow 2=p\]
\[\Rightarrow p=2\]