Question

How do you solve $\dfrac{3}{x} + \dfrac{2}{y} = - \dfrac{5}{6}$ and $\dfrac{1}{x} - \dfrac{1}{y} = - \dfrac{5}{6}$ using substitution?

Answer 1

Hint: This problem deals with solving two equations with two variables. Given a pair of linear equations with variables $x$ and $y$. Now we have to solve these two linear equations with two variables, with the help of a substitution method. It gives two variables, then we know that we need two equations to solve these two variables.

Complete step-by-step solution:
The given two linear equations are $\dfrac{3}{x} + \dfrac{2}{y} = - \dfrac{5}{6}$ and $\dfrac{1}{x} - \dfrac{1}{y} = - \dfrac{5}{6}$
Consider the first equation, as given below:
$ \Rightarrow \;\dfrac{3}{x} + \dfrac{2}{y} = - \dfrac{5}{6}$
Now rearrange the left hand side of the above equation such that the term $\dfrac{3}{x}$ is moved to the right hand side of the equation, as given below:
$ \Rightarrow \dfrac{2}{y} = - \dfrac{5}{6} - \;\dfrac{3}{x}$
Now divide the above equation with 2 on both sides of the equation, as shown below:
$ \Rightarrow \dfrac{1}{y} = - \dfrac{1}{2}\left( {\dfrac{5}{6} + \dfrac{3}{x}} \right)$
Simplifying the above equation, as shown below:
$ \Rightarrow \dfrac{1}{y} = - \left( {\dfrac{{5x + 18}}{{12x}}} \right)$
Now substitute this expression in the second equation$\dfrac{1}{x} - \dfrac{1}{y} = - \dfrac{5}{6}$, as shown below:
\[ \Rightarrow \dfrac{1}{x} - \left( { - \left( {\dfrac{{5x + 18}}{{12x}}} \right)} \right) = - \dfrac{5}{6}\]
\[ \Rightarrow \dfrac{1}{x} + \dfrac{{5x + 18}}{{12x}} = - \dfrac{5}{6}\]
Now simplifying the left hand side of the equation, as shown below:
\[ \Rightarrow \dfrac{{5x + 30}}{{12x}} = - \dfrac{5}{6}\]
Now divide the above equation by 5 on both sides, as given below:
\[ \Rightarrow \dfrac{{x + 6}}{{12x}} = - \dfrac{1}{6}\]
Now cross-multiply to get the value of $x$, as shown below:
\[ \Rightarrow 6\left( {x + 6} \right) = - 12x\]
Now divide the equation by 6, as shown:
\[ \Rightarrow x + 6 = - 2x\]
\[ \Rightarrow x + 2x = - 6\]
Hence the value of $x$ is given by:
\[ \Rightarrow 3x = - 6\]
$\therefore x = - 2$
Now finding the value of $y$, by substituting the value of $x = - 2$, in the second equation $\dfrac{1}{x} - \dfrac{1}{y} = - \dfrac{5}{6}$, as shown below:
$ \Rightarrow - \dfrac{1}{2} - \dfrac{1}{y} = - \dfrac{5}{6}$
Multiplying the equation with -1 and moving the constants on the right side of the equation:
$ \Rightarrow \dfrac{1}{2} + \dfrac{1}{y} = \dfrac{5}{6}$
$ \Rightarrow \dfrac{1}{y} = \dfrac{5}{6} - \dfrac{1}{2}$
Simplifying the constants on the right side of the equation, as shown:
$ \Rightarrow \dfrac{1}{y} = \dfrac{{5 - 3}}{6}$
$ \Rightarrow \dfrac{1}{y} = \dfrac{1}{3}$
Now reciprocating the above equation, gives:
$\therefore y = 3$

The values of x and y are -2 and 3 respectively.

Note: Please note that the given two linear equations with two variables are solved by substitution method, to verify the solutions obtained, we can solve the given two linear equations by the elimination method, which is by adding the two equations, when matched the coefficients of x or y, and then check whether the obtained are correct.