Question

Solve the following systems of equation-
$
  \dfrac{{x + y}}{{xy}} = 2 \\
    \\
  \dfrac{{x - y}}{{xy}} = 6 \\
$

Answer 1

Hint- In such questions, simplify the equations given in the question in order to make it simpler. We will also use the method of substitution to solve this question further.

Complete step-by-step answer:
The equations given in the question are-
$
   \Rightarrow \dfrac{{x + y}}{{xy}} = 2 \\
    \\
   \Rightarrow \dfrac{{x - y}}{{xy}} = 6 \\
$
Simplifying the equations one by one to make the solution simpler-
First equation-
$
   \Rightarrow \dfrac{{x + y}}{{xy}} = 2 \\
    \\
   \Rightarrow \dfrac{x}{{xy}} + \dfrac{y}{{xy}} = 2 \\
    \\
   \Rightarrow \dfrac{1}{y} + \dfrac{1}{x} = 2 \\
$
Mark the above equation as equation 1-
$ \Rightarrow \dfrac{1}{y} + \dfrac{1}{x} = 2$ (equation 1)
Second equation-
$
   \Rightarrow \dfrac{{x - y}}{{xy}} = 6 \\
    \\
   \Rightarrow \dfrac{x}{{xy}} - \dfrac{y}{{xy}} = 6 \\
    \\
   \Rightarrow \dfrac{1}{y} - \dfrac{1}{x} = 6 \\
$
Marking the above equation as equation 2-
$ \Rightarrow \dfrac{1}{y} - \dfrac{1}{x} = 6$ (equation 2)
Now, as we can see that the equations given to us are not linear equations because y is in the denominator in both the equations. So, in order to make the linear, we will-
Let, $\dfrac{1}{x} = v,\dfrac{1}{y} = w$
Now, substituting the value of v and w into equation 1 and equation 2-
$ \Rightarrow v + w = 2$
\[ \Rightarrow - v + w = 6\]
Naming the above equations as equation 3 and 4 respectively, we get-
$ \Rightarrow v + w = 2$ (equation 3)
\[ \Rightarrow - v + w = 6\] (equation 4)
Now, adding equation 3 and equation 4, we get-
$
   \Rightarrow v + w - v + w = 6 + 2 \\
    \\
   \Rightarrow 2w = 8 \\
    \\
   \Rightarrow w = 4 \\
$
Putting this value of w into equation 3, we get the value of v-
$
   \Rightarrow v + w = 2 \\
    \\
   \Rightarrow v + 4 = 2 \\
    \\
   \Rightarrow v = - 2 \\
$
As we already knew $\dfrac{1}{x} = v,\dfrac{1}{y} = w$, putting the values of v and w to get the values of x and y-
$
   \Rightarrow \dfrac{1}{x} = - 2 \\
    \\
   \Rightarrow x = - \dfrac{1}{2} \\
$
And,
$
   \Rightarrow \dfrac{1}{y} = 4 \\
    \\
   \Rightarrow y = \dfrac{1}{4} \\
$
Thus, the values of x and y are $x = - \dfrac{1}{2},y = \dfrac{1}{4}$.

Note: When the equations given are not linear equations, remember to make them linear first or we won’t get even close to the answer. Using the basic method of substitution and elimination will give us the required answer. In this solution, we used the method of substitution.