Question

Solve the following system of equation:
\[\dfrac{1}{{2(x + 2y)}} + \dfrac{5}{{3(3x - 2y)}} = \dfrac{{ - 3}}{2}\], \[\dfrac{5}{{4(x + 2y)}} - \dfrac{3}{{5(3x - 2y)}} = \dfrac{{61}}{{60}}\]

Answer 1

Hint: In these types of questions substitute \[\dfrac{1}{{x + 2y}} = u\] and \[\dfrac{1}{{3x - 2y}} = v\] then use it to find the values of u and v then use the values of u and v to find the values of x and y.

Complete step-by-step answer:
Let \[\dfrac{1}{{x + 2y}} = u\] and \[\dfrac{1}{{3x - 2y}} = v\]
So the equations are
\[\dfrac{1}{2}u + \dfrac{5}{3}v = \dfrac{{ - 3}}{2}\] and \[\dfrac{5}{4}u - \dfrac{3}{5}v = \dfrac{{61}}{{60}}\]
Let \[\dfrac{1}{2}u + \dfrac{5}{3}v = \dfrac{{ - 3}}{2}\] equation 1 and \[\dfrac{5}{4}u - \dfrac{3}{5}v = \dfrac{{61}}{{60}}\] equation 2
Multiplying equation 1 by $\dfrac{3}{5}$ and equation 2 by $\dfrac{5}{3}$
$ \Rightarrow $\[\dfrac{3}{5} \times \dfrac{1}{2}u + \dfrac{3}{5} \times \dfrac{5}{3}v = \dfrac{3}{5} \times \dfrac{{ - 3}}{2}\] and \[\dfrac{5}{3} \times \dfrac{5}{4}u - \dfrac{5}{3} \times \dfrac{3}{5}v = \dfrac{5}{3} \times \dfrac{{61}}{{60}}\]
$ \Rightarrow $\[\dfrac{3}{{10}}u + 1v = \dfrac{{ - 9}}{{10}}\] and \[\dfrac{{25}}{{12}}u - 1v = \dfrac{{305}}{{180}}\]
Adding both the equations with each other
$ \Rightarrow $\[\dfrac{3}{{10}}u + 1v + \dfrac{{25}}{{12}}u - 1v = \dfrac{{305}}{{180}} + \dfrac{{ - 9}}{{10}}\]
$ \Rightarrow $\[u = \dfrac{1}{3}\]
Putting value of u in equation 1
$ \Rightarrow $\[\dfrac{1}{2} \times \dfrac{1}{3} + \dfrac{5}{3}v = \dfrac{{ - 3}}{2}\]
$ \Rightarrow $\[v = - 1\]
Now using the value of u for finding the value of x
\[\dfrac{1}{{x + 2y}} = \dfrac{1}{3}\]
$ \Rightarrow $\[x + 2y = 3\] (Equation 3)
Now using the value of v for finding the value of y
\[\dfrac{1}{{3x - 2y}} = - 1\]
$ \Rightarrow $\[3x - 2y = - 1\] (Equation 4)
Adding equation 3 and 4
$ \Rightarrow $\[3x - 2y + x + 2y = 3 - 1\]
$ \Rightarrow $\[x = \dfrac{1}{2}\]
Putting the value of x in equation 4
$ \Rightarrow $\[3 \times \dfrac{1}{2} - 2y = - 1\]
$ \Rightarrow $\[y = \dfrac{5}{4}\]
Hence the value of x and y are $\dfrac{1}{2},\dfrac{5}{4}$

Note: The equation is in complex form we can simplify the equation by substituting \[\dfrac{1}{{x + 2y}} = u\] and \[\dfrac{1}{{3x - 2y}} = v\] in the equation and then we can compare then we can apply any operation to find the value of u and v since we got the values of variables which we needed to find the value of x and y.