Question

Solve the following system of equations by substitution, elimination and cross multiplication and find the values of x and y.
$
  \dfrac{{2x}}{a} + \dfrac{y}{b} = 2 \\
  \dfrac{x}{a} - \dfrac{y}{b} = 4 \;
 $

Answer 1

Hint: Both the equations in the given system are linear as maximum power of their variables is 1. We can use all the mentioned methods in order to find it solution as per the need for simplification, whichever suits stepwise and obtain the respective values of x and y

Complete step-by-step answer:
The given system of equations is:
$
  \dfrac{{2x}}{a} + \dfrac{y}{b} = 2 \\
  \dfrac{x}{a} - \dfrac{y}{b} = 4 \;
 $
As the highest power of variables is 1, these equations are called as linear equations and we can solve them suing the following methods:
Cross multiplication: If we have two fractions on RHS and LHS of an equation, the numerator on left gets multiplied to the denominator of right and the numerator on right gets multiplied to the denominator of left.
Elimination method: The coefficients of either variable of the equations are made same so that when we subtract that particular variable gets eliminated and we can then find the value of the other one.
Substitution method: When we have the value of one variable, either exact or in terms of the other, we can substitute it so that the complete equation is only in one variable whose value can be easily calculated.
Solving the given system of equations using these methods.
i) Cross multiplying both sides in both the respective equations:
$
  a)\dfrac{{2x}}{a} + \dfrac{y}{b} = 2 \\
  \Rightarrow \dfrac{{2xb + ay}}{{ab}} = 2{\text{ }}\left( {LCM} \right) \\
   \Rightarrow 2xb + ay = 2ab....(1) \\
  b)\dfrac{x}{a} - \dfrac{y}{b} = 4 \\
  \Rightarrow \dfrac{{xb - ay}}{{ab}} = 4{\text{ }}\left( {LCM} \right) \\
   \Rightarrow xb - ay = 4ab....(2) \\
 $
ii) By elimination method:
Multiplying equation 2 with 2, we get:
$
  \Rightarrow 2\left( {xb - ay} \right) = 2 \times 4ab \\
   \Rightarrow 2xb - 2ay = 8ab \\
 $
Subtracting equation (1) from this equation to eliminate xb:
$
  2xb - 2ay - 2xb - ay = 8ab - 2ab \\
  \Rightarrow - 3ay = 6ab \\
  \Rightarrow - ay = 2ab \\
   \Rightarrow y = - 2b \\
 $
iii) Substituting this value of y in equation (2) to find the value of x:
$
  xb - ay = 4ab \\
  \Rightarrow xb - a \times \left( { - 2b} \right) = 4ab \\
  \Rightarrow xb + 2ab = 4ab \\
  \Rightarrow xb = 2ab \\
   \Rightarrow x = 2a \;
 $
Therefore, by substitution, elimination and cross multiplication method, the value of x and y for the given system of equations is 2a and – 2b respectively.
So, the correct answer is “ the value of x and y for the given system of equations is 2a and – 2b respectively”.

Note: whenever we decide which equation the given value is to be substituted, we always choose a less complex equation. In equation (1), there was a coefficient present with the x term but in (2), no such coefficient was present and thus we used that equation for substitution.
In multiplication, two negative signs become positive, two positives remain positive and one positive and one negative also becomes negative.