Question

Solve the following pair of linear equations by the elimination method and the substitution method.
$\dfrac{x}{2} + \dfrac{2}{3}y = - 1\;{\text{and }}x - \dfrac{y}{3} = 3$

Answer 1

Hint- In order to solve the equations first we have to know about elimination method, in the elimination method we either add or subtract the equations to get an equation in one variable and in substitution method we will try to get the value of one variable from an equation then substitute that variable value in other equation.

Complete step-by-step answer:

Given equations are
$
  \dfrac{x}{2} + \dfrac{2}{3}y = - 2\;............................\left( 1 \right) \\

  {\text{ }}x - \dfrac{y}{3} = 3...................................\left( 2 \right) \\

$

Solving by elimination method

First multiply equation (1) by (2), we get

$

  x + \dfrac{4}{3}y = - 2.......................\left( 3 \right) \\

  x - \dfrac{y}{3} = 3......................\left( 2 \right) \\

 $

Now subtracting equation (2) from equation (3), we get

$

   \Rightarrow x + \dfrac{4}{3}y - {\text{ }}x + \dfrac{y}{3} = - 2 - 3 \\

   \Rightarrow \dfrac{5}{3}y = - 5 \\

   \Rightarrow 5y = - 12 \\

   \Rightarrow y = \dfrac{{ - 15}}{5} \\

   \Rightarrow y = - 3 \\

 $

Putting this value in equation (2), we get

$

   \Rightarrow {\text{ }}x - \dfrac{y}{3} = 3 \\

   \Rightarrow x - \dfrac{{\left( { - 3} \right)}}{3} = 3 \\

   \Rightarrow x + 1 = 3 \\

   \Rightarrow x = 2 \\

$

Hence our answer is \[x = 2\] and $y = - 3.$

Solving by substitution method

$ \Rightarrow {\text{ }}x - \dfrac{y}{3} = 3...........\left( 2 \right)$

Add $\dfrac{y}{3}$ both side, we get

$ \Rightarrow x = 3 + \dfrac{y}{3}...........\left( 4 \right)$

Putting this value in equation (1), we get

$

   \Rightarrow \dfrac{x}{2} + \dfrac{2}{3}y = - 2\;............................\left( 1 \right) \\

   \Rightarrow \dfrac{{\left( {3 + \dfrac{y}{3}} \right)}}{2} + \dfrac{{2y}}{3} = - 1 \\

   \Rightarrow \dfrac{3}{2} + \dfrac{y}{6} + \dfrac{{2y}}{3} = - 1 \\

$

Multiplying both sides by (6) we get

$

   \Rightarrow 9 + y + 4y = - 6 \\

   \Rightarrow 5y = - 15 \\

   \Rightarrow y = - 3 \\

 $

Hence the value of $x$ is 2 and $y$ is $ - 3.$


Note- In order to solve these types of questions, we need to learn about how to solve linear equations with single variables and more. As we know that the unique solution of a linear equation can be found only when the number of equations is equal to the number of variables. The linear equations can be solved by following methods: substitution method, elimination method and cross multiplication method.