Question

Find the value of \[p\] and \[q\] for which the system of equations has an infinite number of solutions \[2x + 3y = 7\] and \[(p + q)x + (2p – q)y = 21\] .

Answer 1

Hint: In this question, we need to find the value of \[p\] and \[q\] in the system of equations \[2x + 3y = 7\] and \[(p + q)x + (2p – q)y = 21\] which has infinite number of solutions. First , we can rewrite both the equations such that their right hand sides are equal to zero. Then ,we can know the concept of the consistent equations. Then we need to solve the equations to get the value of \[p\] and \[q\].

Complete answer:
Given two equations \[2x + 3y = 7\] and \[(p + q)x + (2p – q)y = 21\] which have infinitely many solutions.
Now we can rewrite both the equations such that their right hand sides are equal to zero.
That is \[2x + 3y = 7\] can be rewritten as \[2x + 3y – 7 = 0\] and \[(p + q)x + (2p – q)y = 21\] can be rewritten as \[(p + q)x + (2p – q)y – 21 = 0\] .
We can know the concept of the consistent equations.
Let us consider two equations \[a_{1}x + b_{1}y + c_{1} = 0\] and \[a_{2}x + b_{2}y + c_{2} = 0\] .
We can conclude that the lines will have infinitely many solutions when the lines are coincident.
If the two lines are coincident,
That is \[\dfrac{a_{1}}{a_{2}} = \dfrac{b_{1}}{b_{2}} = \dfrac{c_{1}}{c_{2}}\]
Also if the lines are not coincident,
That is \[\dfrac{a_{1}}{a_{2}} = \dfrac{b_{1}}{b_{2}} \neq \dfrac{c_{1}}{c_{2}}\]
Here is the system of equations \[2x + 3y = 7\] and \[(p + q)x + (2p – q)y = 21\] which has an infinite number of solutions.
Here \[a_{1} = 2\ ,\ b_{1} = 3\ ,\ c_{1} = - 7\] and \[a_{2} = \left( p + q \right),b_{2} = \left( 2p – q \right),c_{2} = - 21\].
Now we can use the condition of consistent equations.
We get,
\[\Rightarrow \dfrac{2}{p + q} = \dfrac{3}{2p – q} = - \dfrac{7}{- 21}\]
On simplifying,
We get,
\[\dfrac{2}{p + q} = \dfrac{3}{2p – q} = \dfrac{1}{3}\]
By using this we can easily find out the value of \[p\] and \[q\].
Let us consider \[\dfrac{2}{p + q} = \dfrac{1}{3}\]
On cross multiplying,
We get,
\[\Rightarrow \ 2 \times 3 = p + q\]
On simplifying,
We get,
\[6 = p + q\] ••• (1)
Now let us consider \[\dfrac{3}{2p – q} = \dfrac{1}{3}\]
On cross multiplying,
We get,
\[9 = 2p – q\] ••• (2)
Now we need to solve equations (1) and (2) in order to find the value of \[p\] and \[q\].
On adding equation (1) and (2) ,
We get,
\[3p = 15\]
On dividing both sides by \[3\] ,
We get,
\[\Rightarrow \ p = \dfrac{15}{3}\]
On simplifying,
We get, \[p = 5\]
Now on putting \[p = 5\] in equation (1) ,
We get,
\[6 = 5 + q\]
On subtracting both sides by \[5\] ,
We get,
\[\Rightarrow \ q = 6 – 5\]
On simplifying,
We get,
\[q = 1\]
Thus we get the value of \[p = 5\] and \[q = 1\] .
Final answer :
The given system of equations \[2x + 3y = 7\] and \[(p + q)x + (2p – q)y = 21\] will have an infinite number of solutions if \[p = 5\] and \[q = 1\] .

Note: In order to simplify these types of questions, we should have a strong grip over the consistent equations. Mathematically , a system of linear or nonlinear equations is known as the consistent equations if there is at least one set of values for the unknowns that satisfies all of the equations. Similarly, a system of linear or nonlinear equations is known as the inconsistent equations if there is no set of values for the unknowns that satisfies all of the equations.