How to balance redox equations

Balance the following redox equation using the inspection technique and the oxidation number method:
$$Ag_2O+H_2O_2 \rightarrow Ag+H_2O +O_2$$

SOLUTION:
First: try to balance this chemical equation using the inspection technique (using the principle of conservation of mass). We write down the stoichiometric coefficients:
$${\color{red} a}Ag_2O+{\color{red} b}H_2O_2 \rightarrow {\color{red} c}Ag+{\color{red} d}H_2O +{\color{red} e}O_2$$
For the principle of conservation of mass, we have:
$$\begin{cases}Ag:2a=c\\ H:2b=2d\\ O:a+2b=d+2e\end{cases}$$
We have five variables (from a to e) and only three equation. So we cannot balance correctly the equation only with the inspection technique.

Second: is this a redox equation? Write down all oxidation number.
$$\overset{+1}{Ag_2}\overset{-2}{O}+\overset{+1}{H_2}\overset{-1}{O_2} \rightarrow \overset{0}{Ag}+\overset{+1}{H_2}\overset{-2}{O}+\overset{0}{O_2}$$
As we can see, some atoms change their oxidation number: the Ag and the oxygen. So we can write the oxidating equation for the oxygen and the reducing equation for Ag:
$$\begin{cases}\overset{+1}{Ag_2}+xe^- \rightarrow 2\overset{0}{Ag}\\\\ \overset{-1}{O_2} \rightarrow \overset{0}{O_2}+ye^-\end{cases}$$
Now find the x and y numbers of electrons. There are two electrons in the oxidating and two in the reducting equation because there are two atoms of Ag and two atoms of O. So we have:
$$\begin{cases}\overset{+1}{Ag_2}+2e^- \rightarrow 2\overset{0}{Ag}\\\\ \overset{-1}{O_2} \rightarrow \overset{0}{O_2}+2e^-\end{cases}$$


Third: the electrons are balanced (there are 2 in either the equations). So we can now sum:
$$\begin{cases}\overset{+1}{Ag_2}+2e^- \rightarrow 2\overset{0}{Ag}\\\\ \overset{-1}{O_2} \rightarrow \overset{0}{O_2}+2e^-\end{cases}+$$
$$\rule[0.3cm]{5.5cm}{0.01cm} $$
$$=\quad \overset{+1}{Ag_2}+\overset{-1}{O_2} \rightarrow 2\overset{0}{Ag}+\overset{0}{O_2}$$

So we have found 4 variable values:
$$\begin{cases} a=1\\b=1\\c=2\\e=1\end{cases}$$
From the equation of the hydrogen balancing, we found that $d=1$. We can verify this stoichiometric coefficients substituting the values in the oxygen equation of mass balancing.

So the balanced equation is:
$${\color{red} 1}Ag_2O+{\color{red} 1}H_2O_2 \rightarrow {\color{red} 2}Ag+{\color{red} 1}H_2O +{\color{red} 1}O_2$$
Or:
$$Ag_2O+H_2O_2 \rightarrow {\color{red} 2}Ag+H_2O +O_2$$

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PROBLEM 10:
Balance the following reaction: manganese dioxide + hydrochloric acid produces manganese(II) ions + chlorine gas