Chapter 1.2, Problem 31E

Balancing a chemical reaction. Consider the chemicalreaction
   $a{\text{NO}}_2+b{\text{H}}_{\text{2}}\text{O}\to c{\text{HNO}}_2+d{\text{HNO}}_3,$
where a, b, c, and d are unknown positive integers. Thereaction must be balanced; that is, the number of atomsof each element must be the same before and after thereaction. For example, because the number of oxygenatoms must remain the same, $2a+b=2c+3d.$
While there are many possible values for a, b, c, and dthat balance the reaction, it is customary to use the smallest possible positive integers. Balance this reaction.