Changing the Subject of a Simple Rational Algebraic Fraction

Suppose it required to make  
\[x\]
  the subject of the formula  
\[y=\frac{2x+3}{4x+5}\]

Start by clearing the fraction (multiplying by  
\[4x+5\]
).
\[y(4x+5)=2x+3\]

\[4xy+5y=2x+3\]

We are to make  
\[x\]
  the subject so need to move all the terms containing  
\[x\]
  to the same side of the equals sign, and all other terms to the other side.
\[4xy-2x=-5y+3\]

Now factorise with  
\[x\]

\[x(4x-2)=-5y+3\]

Finally divide by the coefficient of  
\[x, \: 4x-2\]
.
\[x=\frac{-5y+3}{4x-2}\]
.