Question

Find the value of \[Q\] in the following system so that the solution to the system is \[\left\{ {\left( {x,y} \right):x - 3y = 4} \right\}\]?
\[x - 3y = 4\]
\[Qx - 6y = 8\]

Answer 1

Hint: We are given two equations and we need to find value of \[Q\] such that the given system of equations has the solution \[\left\{ {\left( {x,y} \right):x - 3y = 4} \right\}\], which means all those ordered pairs \[\left( {x,y} \right)\] which satisfy the equation \[x - 3y = 4\]. In order to solve this, we need to compare these equations and then find a relation between these coefficients and then find the value of \[Q\].

Complete step-by-step answer:
We are given two equations
\[x - 3y = 4\]
\[Qx - 6y = 8\]
The solution to this system of equations is given \[\left\{ {\left( {x,y} \right):x - 3y = 4} \right\}\].
We see that all the points \[\left( {x,y} \right)\] lying on the line \[x - 3y = 4\] satisfy this line equation.
Let us see this line through a diagram

Now, out of the given two equations, \[x - 3y = 4\] is itself the line.
Now, consider \[Qx - 6y = 8\].
 Comparing it with \[x - 3y = 4\], we see that the coefficient of \[y\] and the constant term in \[Qx - 6y = 8\] is double the coefficient of \[y\] and the constant term in \[x - 3y = 4\].
So, \[Qx - 6y = 8\] will satisfy \[x - 3y = 4\] only if the coefficient of \[x\] in \[Qx - 6y = 8\] is double the coefficient of \[x\] in \[x - 3y = 4\].
Coefficient of \[x\] in \[x - 3y = 4\] is \[1\]
Coefficient of \[x\] in \[Qx - 6y = 8\] is \[Q\]
Now, the coefficient of \[x\] in \[Qx - 6y = 8\] is double the coefficient of \[x\] in \[x - 3y = 4\] implies
\[ \Rightarrow Q = 2 \times 1\]
\[ \Rightarrow Q = 2\]
Hence, we get, the value of \[Q\] in the system \[x - 3y = 4\] and \[Qx - 6y = 8\] for which the system has the solution \[\left\{ {\left( {x,y} \right):x - 3y = 4} \right\}\] is \[2\].

Note: First of all, in such types of questions, we need to learn reading the set. When we are given the solution set, we need to check that the system of equations lies in the given set. We can see it through the diagram. In this we usually get confused between parallel lines and equal lines. Two lines \[ax + by = c\] and \[ux + vy = w\] are said to be parallel if \[\dfrac{a}{u} = \dfrac{b}{y}\] and \[c \ne w\]. Similarly, two lines \[ax + by = c\] and \[ux + vy = w\] are equal if \[\dfrac{a}{u} = \dfrac{b}{y} = \dfrac{c}{w}\].