Question

Solve the following system of equations-
$
  \dfrac{1}{{7x}} + \dfrac{1}{{6y}} = 3 \\
    \\
  \dfrac{1}{{2x}} - \dfrac{1}{{3y}} = 5 \\
$

Answer 1

Hint- The equations above in the question are not linear equations so we will first make them linear using the method shown in the solution below. Then we will use the method of substitution in order to solve the question further.

Complete step-by-step answer:
Let the equations given the question be equation 1 and equation 2-
$ \Rightarrow \dfrac{1}{{7x}} + \dfrac{1}{{6y}} = 3$ (equation 1)
$ \Rightarrow \dfrac{1}{{2x}} - \dfrac{1}{{3y}} = 5$ (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 \dfrac{v}{7} + \dfrac{w}{6} = 3$
$ \Rightarrow \dfrac{v}{2} - \dfrac{w}{3} = 5$
Let the above equation be equation 3 and equation 4-
$ \Rightarrow \dfrac{v}{7} + \dfrac{w}{6} = 3$ (equation 3)
$ \Rightarrow \dfrac{v}{2} - \dfrac{w}{3} = 5$ (equation 4)
Now, equation 3 and equation 4 are linear equations. So, we will solve these equations with the method of substitution, we get-
Consider equation 3-
$
   \Rightarrow \dfrac{v}{7} + \dfrac{w}{6} = 3 \\
    \\
   \Rightarrow \dfrac{1}{7}v = 3 - \dfrac{1}{6}w \\
    \\
   \Rightarrow v = 7\left( {3 - \dfrac{1}{6}w} \right) \\
    \\
   \Rightarrow v = 21 - \dfrac{7}{6}w \\
$
Putting this value in equation 4, we get-
$
   \Rightarrow \dfrac{v}{2} - \dfrac{w}{3} = 5 \\
    \\
   \Rightarrow \dfrac{1}{2}\left( {21 - \dfrac{{7w}}{6}} \right) - \dfrac{1}{3}w = 5 \\
    \\
   \Rightarrow \dfrac{{21}}{2} - \dfrac{{7w}}{{12}} - \dfrac{w}{3} = 5 \\
    \\
   \Rightarrow \dfrac{{21}}{2} - \dfrac{{7w - 4w}}{{12}} = 5 \\
    \\
   \Rightarrow \dfrac{{21}}{2} - \dfrac{{11w}}{{12}} = 5 \\
    \\
   \Rightarrow \dfrac{{11w}}{{12}} = \dfrac{{21}}{2} - 5 = \dfrac{{11}}{2} \\
    \\
   \Rightarrow \dfrac{{11w}}{{12}} = \dfrac{{11}}{2} \\
    \\
   \Rightarrow w = 6 \\
$
Substituting the value w in $v = 21 - \dfrac{7}{6}w$, we get-
$
   \Rightarrow v = 21 - \dfrac{7}{6}w \\
    \\
   \Rightarrow v = 21 - \dfrac{7}{6}.6 \\
    \\
   \Rightarrow v = 14 \\
$
As we already know $\dfrac{1}{x} = v,\dfrac{1}{y} = w$, substituting these values of v and w into this equation-
$
   \Rightarrow \dfrac{1}{x} = v \Rightarrow \dfrac{1}{x} = 14 \Rightarrow x = \dfrac{1}{{14}} \\
    \\
   \Rightarrow \dfrac{1}{y} = w \Rightarrow \dfrac{1}{y} = 6 \Rightarrow y = \dfrac{1}{6} \\
$
Hence, the value of x and y is $x = \dfrac{1}{{14}},y = \dfrac{1}{6}$.

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 our required answer. In this solution, we used the method of substitution.