A two-digit number is seven times the sum of its digits and also equal to $12$ less than three times the product of its digits. Find the number.

Step 1: Finding the equations

Let the units digit be $y$, so the tens digit will be $x$.

The number is $10x+y$.

According to the given statement in the question, the equation will be,

$10x+y=7(x+y)$

On simplifying further,

$$\begin{gathered} \Rightarrow 10x-7x-7y+y=0 \\ \Rightarrow 3x-6y=0-----\left(i\right) \\ \because 10x+y=3xy-12 \\ \Rightarrow 10x-3xy+y+12=0-----\left(ii\right) \end{gathered}$$

Step 2: Putting the value of $x=2y$ from equation $\left(i\right)$ in $\left(ii\right)$

$$\begin{gathered} \Rightarrow 20y-6y^2+y+12=0 \\ \Rightarrow 21y-6y^2+12=0 \\ \Rightarrow 3(2y^2-7y-4)=0 \\ \Rightarrow 2y^2-7y-4=0 \end{gathered}$$

Step 3: Apply the factorization method

$$\begin{gathered} \Rightarrow 2y^2-7y-4=0 \\ \Rightarrow 2y^2-8y+y-4=0 \\ \Rightarrow 2y(y-4)+1(y-4)=0 \\ \Rightarrow (2y+1)(y-4)=0 \end{gathered}$$

After simplifying,

$y=-\frac{1}{2}or4$

Ignoring the negative fractional value as it is a digit of a two-digit number.

$\therefore y=4$

Then, the value of $x$ will be $x=2\times 4$ from the equation $\left(i\right)$.

$x=8$

Put the value of $x$ and $y$ in $10x+y$.

$$\begin{gathered} \Rightarrow 10x+y=10\left(8\right)+4 \\ \Rightarrow 10x+y=80+4 \\ \Rightarrow 10x+y=84 \end{gathered}$$

Therefore, the number is $84$.