Find the solution set for each equation if the replacement set is $\mathrm{\{}\mathrm{11}\mathrm{,}\mathrm{12}\mathrm{,}\mathrm{13}\mathrm{,}\mathrm{14}\mathrm{,}\mathrm{15}\mathrm{\}}$.

$(k-8)12=84$

Set:- A set is a collection of objects or numbers which is shown using braces.

Replacement Set: A set from which the values of the variable of a given equation are chosen, is known as a replacement set.

Solution Set: A solution set is a set of elements chosen from the replacement set which satisfy the equation or make an open sentence true.

To find the solution set from the replacement set,

We have to follow the given steps:

1. Take value for n from the replacement set.
   
   
2. Solve the equation for k.
   
   
3. After solving if both sides have equal value.
   Take k as an element of the solution set.

Let’s substitute the values of the replacement set one by one in the equation.

Take k = 11 from replacement set and put in given equation(open sentence).

$\begin{array}{l}(k-8)12=84\\ \Rightarrow (11-8)12=84\\ \Rightarrow \left(3\right)12=84\\ \Rightarrow \text{36}=84\end{array}$

After solving, we get 36 = 84, which is False.

Hence k = 11 doesn’t belong to the solution set.

Take k = 12 from the replacement set and put it in the given equation.

$\begin{array}{l}(k-8)12=84\\ \Rightarrow (12-8)12=84\\ \Rightarrow \left(4\right)12=84\\ \Rightarrow \text{48}=84\end{array}$

After solving, we get 48 = 84, which is False.

Hence k = 12 doesn’t belong to the solution set.

Take k = 13 from the replacement set and put it in the equation.

$\begin{array}{l}(k-8)12=84\\ \Rightarrow (13-8)12=84\\ \Rightarrow \left(5\right)12=84\\ \Rightarrow \text{60}=84\end{array}$

After solving, we get 60 = 84, which is False.

Hence k = 13 doesn’t belong to the solution set.

Take k = 14 from the replacement set and put it in the equation.

$\begin{array}{l}(k-8)12=84\\ \Rightarrow (14-8)12=84\\ \Rightarrow \left(6\right)12=84\\ \Rightarrow \text{72}=84\end{array}$

After solving, we get 72 = 84, which is False.

Hence k = 14 doesn’t belong to the solution set.

Take k = 15 from the replacement set and put it in the equation.

$\begin{array}{l}(k-8)12=84\\ \Rightarrow (15-8)12=84\\ \Rightarrow \left(7\right)12=84\\ \Rightarrow \text{84}=84\end{array}$

After solving, we get 84 = 84, which is True.

Hence n = 15 doesn’t belong to the solution set.

We can observe that only the value for k = 15 belongs to the solution set.

Therefore, the solution set is {15}.