Question

Solve the following simultaneous equations.

\[\frac{7}{2x + 1} + \frac{13}{y + 2} = 27 ; \frac{13}{2x + 1} + \frac{7}{y + 2} = 33\]

Answer 1

\[\frac{7}{2x + 1} + \frac{13}{y + 2} = 27 ; \frac{13}{2x + 1} + \frac{7}{y + 2} = 33\]

 

\[\frac{1}{2x + 1} = u\text{ and }\frac{1}{y + 2} = v\]

 

\[7u + 13v = 27 . . . . . \left( I \right)\]

\[13u + 7v = 33 . . . . . \left( II \right)\]

 

(I) + (II)

\[20u + 20v = 60\]
\[u + v = 3 . . . . . \left( III \right)\]

 

(II) − (I)

\[6u - 6v = 6 \]
\[u - v = 1 . . . . . \left( IV \right)\]

 

(III) + (IV)

\[2u = 4\]
\[ \Rightarrow u = 2\]

 

Putting the value of u in (IV) 
\[2 - v = 1\]

\[ \Rightarrow v = 1\]
\[\frac{1}{2x + 1} = u = 2 \]

\[ \Rightarrow 2x + 1 = \frac{1}{2}\]
\[ \Rightarrow x = \frac{- 1}{4}\]
\[\text{ and }\frac{1}{y + 2} = v = 1\]
\[ \Rightarrow y + 2 = 1\]
\[ \Rightarrow y = - 1\]
\[\left( x, y \right) = \left( \frac{- 1}{4}, - 1 \right)\]