Question

Solve the following systems of equations:
$
  \dfrac{x}{7} + \dfrac{y}{3} = 5 \\
  \dfrac{x}{2} - \dfrac{y}{9} = 6 \\
 $

Answer 1

Hint – Write both the equations in a simplified form as
$
  \dfrac{x}{7} + \dfrac{y}{3} = 5 \\
   \Rightarrow 3x + 7y - 105 = 0 \\
$
and
$
  \dfrac{x}{2} - \dfrac{y}{9} = 6 \\
   \Rightarrow 9x - 2y - 108 = 0 \\
$
and then multiply by 3 in the first equation and by adding or subtracting the two equations find the values of x and y.

Complete step-by-step answer:
We have the equations-
$
  \dfrac{x}{7} + \dfrac{y}{3} = 5 - (1) \\
  \dfrac{x}{2} - \dfrac{y}{9} = 6 - (2) \\
$
Now taking equation (1) and simplifying it we get-
$
  \dfrac{x}{7} + \dfrac{y}{3} = 5 \\
   \Rightarrow 3x + 7y - 105 = 0 - (3) \\
 $
And then simplifying equation (2) we get-
$
  \dfrac{x}{2} - \dfrac{y}{9} = 6 \\
   \Rightarrow 9x - 2y - 108 = 0 - (4) \\
 $
Multiplying by 3 in equation (3), we get-
\[
   \Rightarrow 3(3x + 7y - 105 = 0) \\
   \Rightarrow 9x + 21y - 315 = 0 - (5) \\
\]
Subtracting equation (5) from (4), we get-
$
  9x - 2y - 108 - (9x + 21y - 315) = 0 \\
   \Rightarrow 9x - 2y - 108 - 9x - 21y + 315 = 0 \\
   \Rightarrow - 23y + 207 = 0 \\
   \Rightarrow y = \dfrac{{207}}{{23}} = 9 \\
$
Now putting the value of y in equation 3 we get-
$
  3x + 7 \times 9 - 105 = 0 \\
   \Rightarrow 3x + 63 - 105 = 0 \\
   \Rightarrow 3x - 42 = 0 \\
   \Rightarrow x = \dfrac{{42}}{3} = 14 \\
$
Therefore, the values of x and y are 14 and 9 respectively.

Note - Whenever such types of questions appear then always write the equations given in the question and then simplify them to the simplest form. Then by adding or subtracting try to find the value of x or y and then by using the value of either x or y, find the other unknown variable.