# Solve the simultaneous equation:

$ \dfrac{1}{x} + \dfrac{1}{y} = 12,\dfrac{3}{x} - \dfrac{2}{y} = 1 $ .

Last updated date: 15th Mar 2023

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Answer

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**Hint**: To solve a pair of equations that are reducible to linear form, find the expressions that repeat in both the equations.

Give them a simpler form, say $ x $ and $ y $ . Solve the new pair of linear equations for the new variables.

**:**

__Complete step-by-step answer__Given equations are,

$ \Rightarrow $ $ \dfrac{1}{x} + \dfrac{1}{y} = 12,\dfrac{3}{x} - \dfrac{2}{y} = 1 $

Substituting $ \dfrac{1}{x} = a $ and $ \dfrac{1}{y} = b $ in both above equations.

$ \Rightarrow a + b = 12 $ _ _ _ _ _ _ _ _ _ _ $ \left( 1 \right) $

$ \Rightarrow 3a - 2b = 1 $ _ _ _ _ _ _ _ _ _ _ $ \left( 2 \right) $

Multiply equation $ \left( 1 \right) $ by $ 2 $ and solving with $ \left( 2 \right) $ ,

$ \Rightarrow 2a + 2b = 24 $ _ _ _ _ _ _ _ _ _ _ _ $ \left( 3 \right) $

$ \Rightarrow 3a - 2b = 1 $

________________

$ \Rightarrow 5a = 25 $

$ \Rightarrow a = \dfrac{{25}}{5} = 5 $

Substituting $ a = 5 $ in equation $ \left( 1 \right) $ ,

$

\Rightarrow a + b = 12 \\

\Rightarrow 5 + b = 12 \\

\Rightarrow b = 12 - 5 = 7 \;

$

Therefore,

$

\Rightarrow \dfrac{1}{x} = a \\

\Rightarrow x = \dfrac{1}{5}\;

$

$

\Rightarrow \dfrac{1}{y} = b \\

\Rightarrow y = \dfrac{1}{7} \;

$

**So, the correct answer is “$x = \dfrac{1}{5}$ and $y = \dfrac{1}{7}$”.**

**Note**: $ \Rightarrow $ Equations in which the powers of all the variables involved are one are called linear equations. The degree of a linear equation is always one.

$ \Rightarrow $ Identify unknown quantities and denote them by variables.

$ \Rightarrow $ Represent the relationships between quantities in a mathematical form, replacing the unknowns with variables.

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