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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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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.

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.