Solve the equation and check the solution.

$|y-5|-2=10$

For any real numbers $a,b$, where $b\ge 0$, if $\left|a\right|=b$ then $a=b$ or $a=-b$.

Consider the equation $|y-5|-2=10$.

Add 2 on both sides of the equation

$\begin{array}{l}|y-5|-2=10\\ |y-5|-2+2=10+2\\ |y-5|=12\end{array}$

Use the concept of absolute value equation as follows:

$y-5=-12$ or $y-5=12$.

Case 1. Simplify $y-5=-12$.

Add 12 on both sides of the equation and simplify as follows:

$\begin{array}{l}y-5=-12\\ y-5+12=-12+12\\ y+7=0\\ y=-7\end{array}$

Case 2. Simplify $y-5=12$.

Subtract 12 from both sides of the equation and simplify as follows:

$\begin{array}{l}y-5=12\\ y-5-12=12-12\\ y-17=0\\ y=17\end{array}$

Therefore, the solution is $y=-7,17$.

Case 1. Substitute $y=-7$in the equation $|y-5|-2=10$and simplify as follows:

$\begin{array}{l}|y-5|-2=10\\ |-7-5|-2=10\\ |-12|-2=10\\ 12-2=10\\ 10=10\end{array}$

Case 2. Substitute $y=17$in the equation $|y-5|-2=10$and simplify as follows:

$\begin{array}{l}|y-5|-2=10\\ |17-5|-2=10\\ \left|12\right|-2=10\\ 12-2=10\\ 10=10\end{array}$

Since the left-hand side of the equation is equal to the right-hand side of the equation in both the cases therefore, the solution is $y=-7,17$.