You follow a systematic procedure to balance the equation.

Start with the given equation:

#"Zn" + "Pb(NO"_3)_2 → "Pb" + "Zn(NO"_3)_2#

A method that often works is to balance everything other than #"O"# and #"H"# first, then balance #"O"#, and finally balance #"H"#.

Another useful procedure is to start with what looks like the most complicated formula.

The most complicated formula looks like #"Pb(NO"_3)_2# or #"Zn(NO"_3)_2#.

Let's choose #"Pb(NO"_3)_2#. We put a 1 in front of it to remind ourselves that the coefficient is now fixed.

We start with

#"Zn" + color(red)(1) "Pb(NO"_3)_2 → "Pb" + "Zn(NO"_3)_2#

**Balance #"Pb"#:**

We have fixed 1 #"Pb"# on the left, so we need 1 #"Pb"# on the right. We put a 1 in front of the #"Pb"#.

#"Zn" + color(red)(1) "Pb(NO"_3)_2 → color(green)(1) "Pb" + "Zn(NO"_3)_2#

**Balance #"N"#:**

We have fixed 2 #"N"# on the left, so we need 2 #"N"# on the right. We put a 1 in front of the #"Zn(NO"_3)_2#.

#"Zn" + color(red)(1) "Pb(NO"_3)_2 → color(green)(1) "Pb" + color(orange)(1) "Zn(NO"_3)_2#

**Balance #"Zn"#:**

We have fixed 1 #"Zn"# atom on the right so we need 1 #"Zn"# atom on the left. We put a 1 in front of the #"Zn"#.

#color(blue)(1)"Zn" + color(red)(1) "Pb(NO"_3)_2 → color(green)(1) "Pb" + color(orange)(1) "Zn(NO"_3)_2#

Every formula now has a fixed coefficient. We should have a balanced equation.

Let’s check:

#"Element"color(white)(m)"Left-hand side"color(white)(m)"Right-hand side"#

#color(white)(ml)"Zn"color(white)(mmmmmll)1color(white)(mmmmmmmll)1#

#color(white)(ml)"Pb"color(white)(mmmmmll)1color(white)(mmmmmmmll)1#

#color(white)(ml)"N"color(white)(mmmmmml)2color(white)(mmmmmmmll)2#

#color(white)(ml)"O"color(white)(mmmmmml)6color(white)(mmmmmmmll)6#

All atoms balance. The balanced equation is

#color(red)("Zn" + "Pb"("NO"_3)_2 → "Pb" + "Zn"("NO"_3)_2#

Surprise! The original equation was already balanced.