Question

Solve: $ 3(2x + y) = 7xy $ ; $ 3(x + 3y) = 11xy $ using elimination method.

Answer 1

Hint:
Here we need to just eliminate one variable from the above two equations. It can be done by making the coefficient of any one variable equal in both the equations and then we can subtract them to eliminate one variable. Then it can easily be solved.

Complete step by step solution:
Here we are given the two equations and we need to solve them using the elimination method.
First we will make one of the variable’s coefficient equal in both the equations and then we can subtract them to eliminate one variable. Then it can easily be solved.
So we can write the equations as
 $ 3(2x + y) = 7xy $
 $ 3(x + 3y) = 11xy $
We can simplify the above two equations by dividing both the sides by $ xy $
Then we get:
 $
  \dfrac{{3(2x + y)}}{{xy}} = \dfrac{{7xy}}{{xy}} \\
  \dfrac{{6x}}{{xy}} + \dfrac{{3y}}{{xy}} = 7 \\
  $
 $ \dfrac{6}{y} + \dfrac{3}{x} = 7 $ $ - - - (1) $
Now similarly the second given equation becomes:
 $
  \dfrac{{3(x + 3y)}}{{xy}} = \dfrac{{11xy}}{{xy}} \\
  \dfrac{{3x}}{{xy}} + \dfrac{{9y}}{{xy}} = 11 \\
  $
 $ \dfrac{3}{y} + \dfrac{9}{x} = 11 $ $ - - - (2) $
To make our calculations easier let us suppose $ \dfrac{1}{x} = u{\text{ and }}\dfrac{1}{y} = v $
Now our equations become:
 $ 6v + 3u = 7 $ $ - - - - - (3) $
 $ 3v + 9u = 11 $ $ - - - - (4) $
Now we need to solve these equations first:
So we can make the coefficient of $ v $ equal in both the equation and then subtract the both equations and get the value of $ u $
So applying
 $ {\text{equation}}(3) - 2 \times {\text{equation}}(4) $
We get:
 $
  6v + 3u - 2(3v + 9u) = 7 - 2(11) \\
  6v + 3u - 6v - 18u = 7 - 22 \\
  3u - 18u = - 15 \\
   - 15u = - 15 \\
  u = 1 \\
  $
Hence we get $ u = 1 $
Now putting $ u = 1 $ in the equation (3), we get:
 $
  6v + 3u = 7 \\
  6v + 3(1) = 7 \\
  6v = 7 - 3 \\
  6v = 4 \\
  v = \dfrac{4}{6} = \dfrac{2}{3} \\
  $
So we get $ v = \dfrac{2}{3} $
But we need to find the value of $ x,y $
Hence substituting the value of $ u{\text{ and }}v $
 $
  u = \dfrac{1}{x} = 1 \\
  x = 1 \\
  $
 $
  v = \dfrac{1}{y} = \dfrac{2}{3} \\
  2y = 3 \\
  y = \dfrac{3}{2} \\
  $
Hence we get the values of x and y by the use of the elimination method where we eliminated one variable and got another value by the substitution of the found value. In this way we can easily solve the problem with the help of the elimination method.

So we got the variables as $ x = 1,y = \dfrac{3}{2} $

Note:
Here a student can try this problem by eliminating another variable. We eliminated the variable $ v $ first but if we eliminate the variable $ u $ first then also we will get the same result.
One needs to know how we will be able to make the coefficients of any one variable equal in both the equations so that our elimination of one variable becomes easier.