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# Solve the given equation $\dfrac{{{\text{3x}}}}{2} - \dfrac{{{\text{5y}}}}{3} = - 2$, $\dfrac{{\text{x}}}{3} + \dfrac{{\text{y}}}{2} = \dfrac{{13}}{6}$

Last updated date: 17th Mar 2023
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Hint – Using the given equations in the question we transform one equation such that the variable x is in terms of y. Then substitute y in the other equation. Then solve for the value of y and substitute for x.

Given: $\dfrac{{{\text{3x}}}}{2} - \dfrac{{{\text{5y}}}}{3} = - 2$ --- (1)
$\dfrac{{\text{x}}}{3} + \dfrac{{\text{y}}}{2} = \dfrac{{13}}{6}$ ---- (2)

Rewriting (1),
$\dfrac{{{\text{3x}}}}{2} = - 2 + \dfrac{{{\text{5y}}}}{3} \\ \Rightarrow \dfrac{{{\text{3x}}}}{2} = - \dfrac{{{\text{ - 6 + 5y}}}}{3} \\ \Rightarrow {\text{3x}} = - \dfrac{{{\text{ - 12 + 10y}}}}{3} \\ \Rightarrow {\text{x}} = - \dfrac{{{\text{ - 12 + 10y}}}}{{\text{9}}}{\text{ - - - - - - - - }}\left( 3 \right) \\$

Substituting (3) in (2)
$\dfrac{{\left( { - 12 + 10{\text{y}}} \right)}}{9} \times \dfrac{1}{3} + \dfrac{{\text{y}}}{2} = \dfrac{{13}}{6} \\ \Rightarrow \dfrac{{\left( { - 12 + 10{\text{y}}} \right)}}{{27}} + \dfrac{{\text{y}}}{2} = \dfrac{{13}}{6} \\ \Rightarrow \dfrac{{\left( { - 24 + 20{\text{y + 27y}}} \right)}}{{54}} = \dfrac{{13}}{6} \\ \Rightarrow \dfrac{{\left( { - 24 + 47{\text{y}}} \right)}}{{54}} = \dfrac{{13}}{6} \\ \Rightarrow \dfrac{{\left( { - 24 + 47{\text{y}}} \right)}}{9} = 13 \\ \Rightarrow \left( { - 24 + 47{\text{y}}} \right) = 9 \times 13 = 117 \\ \Rightarrow 47{\text{y = 117 + 24}} \\ \Rightarrow {\text{47y = 141}} \\ \Rightarrow {\text{y = 3}} \\$

Substitute value of y in (3) to find value of x

x = $\dfrac{{\left( { - 12 + 10{\text{y}}} \right)}}{9}$
$\Rightarrow {\text{x = }}\dfrac{{\left( { - 12 + 10 \times 3} \right)}}{9} \\ \Rightarrow {\text{x = }}\dfrac{{\left( { - 12 + 30} \right)}}{9} \\ \Rightarrow {\text{x = }}\dfrac{{18}}{9} \\ \Rightarrow {\text{x = 2}} \\$
Hence, (x, y) = (2, 3)

Note – This is a question which is of the type in which there are 2 equations and 2 variables. The key is to transform one of the equations such that we have one variable in terms of another. Then the other equation reduces into a single variable equation and becomes easier to solve. On finding the value of one variable the other can be found simply by substituting.