A two digit number is seven …

Let digit at unit's place be x and digit at ten's place be y 
Therefore, Number = 10y + x
According to given situation we have,

     10y + x = 7(x + y)
{tex}\Rightarrow{/tex}10y + x = 7x + 7y
{tex}\Rightarrow{/tex}6x = 3y
{tex}\Rightarrow{/tex}y =2x   .......................(i)
Also 10y + x = 3xy - 12
{tex}\Rightarrow{/tex}10 {tex}\times{/tex} 2x + x = 3x{tex}\cdot{/tex} 2x - 12
{tex}\Rightarrow{/tex}6x² - 21x -12 = 0 {tex}\Rightarrow{/tex} 2x² - 7x - 4 = 0
{tex}\Rightarrow{/tex}2x² - 8x + x - 4 = 0

Factorize above quadratic equation we get,
     ​​​​​​2x(x - 4) + 1(x - 4) = 0
{tex}\Rightarrow{/tex}(x - 4)(2x + 1) = 0
{tex}\Rightarrow{/tex}x =4 or x = -{tex}\frac{1}{2}{/tex}(rejected)
When x = 4, y = 2 {tex}\times{/tex}4 = 8
{tex}\therefore{/tex}Number is 10{tex}\times{/tex}8 + 4 = 84.