Question

Find the value of a for which the following system of equations will have an infinite number of solutions:
 $ 2x - y = 8 $ and $ 6x - 3y = 4a $

Answer 1

Hint: In the given question, we are required to find the value of the variable ‘a’ such that the given two equations in two variables have an infinite number of solutions. We must know the condition for the system of linear equations in two variables to have infinitely many solutions to solve such questions. We first convert the given two equations into standard forms of linear equations in two variables and then fulfill the requirements for having infinite solutions for a system of equations.

Complete step-by-step answer:
There are three types of systems of linear equations in two variables. One is the system of linear equations in two variables where there is no solution of the equations. Both the equations represent parallel lines in such a case.
Second is the system of linear equations in two variables where only one unique solution of the equations is present. Such equations represent two intersecting straight lines on the Cartesian plane.
Third is the system of linear equations in two variables where there are infinitely many solutions of the equations. Such equations represent the same line on the Cartesian plane.
The equations of the given equations are: $ 2x - y = 8 $ and $ 6x - 3y = 4a $ .
So, for the equations to have infinitely many solutions, the equations must represent the same line on the Cartesian plane. So, the equation of the straight line $ 6x - 3y = 4a $ must be a rational multiple of the equation $ 2x - y = 8 $ .
Now, we know that the general form of a linear equation in two variables is $ ax + by + c = 0 $
So, if a system has infinite number of solutions, then,
 $ \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}} $
Here, for the equation $ 2x - y - 8 = 0 - - - - \left( 1 \right) $ , we have,
 $ {a_1} = 2 $ , $ {b_1} = - 1 $ and $ {c_1} = - 8 $ .
Also, for the equation $ 6x - 3y - 4a = 0 - - - - \left( 2 \right) $ , we have,
 $ {a_2} = 6 $ , $ {b_2} = - 3 $ and $ {c_2} = - 4a $ .
Now, $ \dfrac{{{a_1}}}{{{a_2}}} = \dfrac{{{b_1}}}{{{b_2}}} = \dfrac{{{c_1}}}{{{c_2}}} $
Substituting the values, we get,
 $ \Rightarrow \dfrac{2}{6} = \dfrac{{ - 1}}{{ - 3}} = \dfrac{{ - 8}}{{ - 4a}} $
Cancelling the common factors in numerator and denominator, we get,
 $ \Rightarrow \dfrac{{ - 8}}{{ - 4a}} = \dfrac{1}{3} $
Cross multiplying the terms of equation, we get,
 $ \Rightarrow \dfrac{2}{a} = \dfrac{1}{3} $
 $ \Rightarrow a = 6 $
So, we get the value of a as $ 6 $ .
So, the correct answer is “6”.

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. One should know about the transposition rule so as to shift terms from one side of the equation to another and solve the equation. General form of a linear equation in two variables must be remembered.