Question

Solve the pair of equations
\[ \dfrac{3}{{2x}} + \dfrac{2}{{3y}} = \dfrac{{ - 1}}{3} \\
  \dfrac{3}{{4x}} + \dfrac{1}{{2y}} = \dfrac{{ - 1}}{8} \]

Answer 1

Hint: We will use the method of elimination by equating the coefficients of either \[x\] or \[y\]to find the desired solution that satisfies both the equations. Eliminate either of the variables and substitute it in one of the equations to get the other variable.

Complete step-by-step solution:
We are given with the equations,
\[\dfrac{3}{{2x}} + \dfrac{2}{{3y}} = \dfrac{{ - 1}}{3}\]--------- \[(1)\]
\[\dfrac{3}{{4x}} + \dfrac{1}{{2y}} = \dfrac{{ - 1}}{8}\]--------- \[(2)\]
Now multiplying equation \[(1)\] with \[\dfrac{1}{2}\] and equation \[(2)\] with \[1\] and subtracting we get,
\[\dfrac{1}{2} \times (\dfrac{3}{{2x}} + \dfrac{2}{{3y}} = \dfrac{{ - 1}}{3})\]
\[ \Rightarrow \dfrac{3}{{4x}} + \dfrac{1}{{3y}} = \dfrac{{ - 1}}{6}\]--------\[(3)\]
\[\dfrac{3}{{4x}} + \dfrac{1}{{2y}} = \dfrac{{ - 1}}{8}\]----------\[(4)\]
Now subtracting we get,
\[
   \Rightarrow \dfrac{3}{{4x}} + \dfrac{1}{{3y}} + \dfrac{1}{6} - (\dfrac{3}{{4x}} + \dfrac{1}{{2y}} + \dfrac{1}{8}) = 0 \\
   \Rightarrow \dfrac{1}{{3y}} - \dfrac{1}{{2y}} = \dfrac{1}{8} - \dfrac{1}{6} \\
   \Rightarrow \dfrac{1}{y}(\dfrac{1}{3} - \dfrac{1}{2}) = \dfrac{6}{{8 \times 6}} - \dfrac{8}{{6 \times 8}} \\
   \Rightarrow \dfrac{1}{y}(\dfrac{2}{{3 \times 2}} - \dfrac{3}{{2 \times 3}}) = \dfrac{{ - 2}}{{48}} \\
   \Rightarrow \dfrac{{ - 1}}{{6y}} = \dfrac{{ - 2}}{{48}} \\
   \Rightarrow \dfrac{1}{{6y}} = \dfrac{1}{{24}} \\
   \Rightarrow 6y = 24 \\
   \Rightarrow y = 4 \]
Now substituting \[y = 4\]in equation\[(1)\]we get,
\[
  \dfrac{3}{{2x}} + \dfrac{2}{{3.4}} = \dfrac{{ - 1}}{3} \\
   \Rightarrow \dfrac{3}{{2x}} + \dfrac{2}{{12}} = \dfrac{{ - 1}}{3} \\
   \Rightarrow \dfrac{3}{{2x}} = - \dfrac{2}{{12}} - \dfrac{1}{3} \\
   \Rightarrow \dfrac{3}{{2x}} = - (\dfrac{1}{6} + \dfrac{1}{3}) \\
   \Rightarrow \dfrac{3}{{2x}} = - (\dfrac{1}{6} + \dfrac{2}{6}) \\
   \Rightarrow \dfrac{3}{{2x}} = - \dfrac{3}{6} \\
   \Rightarrow 2x = - 6 \\
   \Rightarrow x = - 3 \]
Therefore the solution to the equations is \[( - 3,4)\]
Additional information: A linear equation is an equation that is written for two exceptional variables. This equation may be a linear aggregate of those variables, and a steady state can be present. Surprisingly, while any linear equation is plotted on a graph, it will necessarily produce an instantaneous line - as a result they are called: Linear equations. Linear Equations are an extensive sort of equations altogether. There may be linear equations in one variable, linear equations in two variables, and so on. In every equation, one issue stays regular: The highest and the best diploma of all variables in the equation should be \[1\]. Other than that, constants \[0\] diploma variables can be there.

Note: It is very important that we know how to calculate the LCM of the numbers if needed and then take one variable from either of the variables to calculate the LCM of the coefficients and then multiply each of them to eliminate either of the variables and then calculate the value of the other variable using substitution method.