The system of nonlinear equations from the given statement and find the numbers

Calculation:
It is given that the sum of two numbers is 20 and their product is 96.
Consider the first part of the sentence as "sum of two numbers is 20" and the second part as "product is 96".
Step 1: Assume the numbers to be x and y.
Step 2: Model the given conditions into a system of equations.
From the first part, the equation is modeled as ${\displaystyle x+y=20}$ and from the second part, the equation is ${\displaystyle xy=96}$.
Thus, the system of equations becomes $\{\begin{array}{l}x+y=20\\ xy=96\end{array}$
Step 3: Modify the equation ${\displaystyle xy=96}$ as ${\displaystyle y=\frac{96}{x}}$.
Substitute ${\displaystyle y=\frac{96}{x}}$ in the equation ${\displaystyle x+y=20}$ and obtain the quadratic equation as,
${\displaystyle x+\left(\frac{96}{x}\right)=20}$
${\displaystyle x^2+96=20x}$ (multiplied by x on both sides)
${\displaystyle x^2-20x+96=0}$
Solve the equation ${\displaystyle x^2-20x+96=0}$ and obtain the value of x as follows.
${\displaystyle x^2-20x+96=0}$
${\displaystyle x^2-12x-8x+96=0}$
${\displaystyle x(x-12)-8(x-6)=0}$
${\displaystyle (x-12)(x-8)=0}$
On further simplification gives,
${\displaystyle x-12=0}$ or ${\displaystyle x-8-0}$
${\displaystyle x=12}$ or ${\displaystyle x=8}$
Substitute ${\displaystyle x=12}$ in the equation ${\displaystyle xy=96}$ and obtain the value of y as follows.
${\displaystyle xy=96}$
${\displaystyle \left(12\right)y=96}$
${\displaystyle y=\frac{96}{12}}$
${\displaystyle y=8}$
Substitute ${\displaystyle x=8}$ in the equation ${\displaystyle xy=96}$ and obtain the value of y as follows.
${\displaystyle xy=96}$
${\displaystyle \left(8\right)y=96}$
${\displaystyle y=\frac{96}{8}}$
${\displaystyle y=12}$
Thus, for ${\displaystyle x=12,y=8}$, and for ${\displaystyle x=8,y=12}$.
Hence, the solution set is ${\displaystyle \{(8,12),(12,8)\}}$.
Step 4: Check the result, by substituting the obtained solutions in the given original equations ${\displaystyle x+y=20}$ and ${\displaystyle xy=96}$.
Substitute (8,12) in the given system and check.
${\displaystyle \left(8\right)+\left(12\right)=20}$
${\displaystyle 20=20}$
${\displaystyle \left(8\right)\left(12\right)=96}$
${\displaystyle 96=96}$
Substitute (12,8) in the given system and check.
${\displaystyle \left(12\right)+\left(8\right)=20}$
${\displaystyle 20=20}$
${\displaystyle \left(12\right)\left(8\right)=96}$
${\displaystyle 96=96}$
Therefore, the two numbers are 12 and 8 or 8 and 12.