Answer
Numbers are $19$ and $27$.
Work Step by Step
Let first number be $x$ and second number be $y$.
First number is $8$ less than a second number.
$x = y - 8$
$x - y = -8$ Equation $(1)$
Twice the first number is $11$ more than the second number.
$2x = y + 11$
$2x - y = 11$ Equation $(2)$
Solving by elimination method,
Multiply Equation (1) by $-1$
$-x +y = 8$ Equation $(3)$
Add Equation $(3)$ and Equation $(2)$ to eliminate $y$.
$-x+2x+y-y=8+11$
$x = 19$
Substituting $x$ value in Equation $(1)$, to get $y$
$x - y = -8$
$x+8 = y$
$19 + 8 = y$
$y = 27$
First number is $19$ and Second number is $27$.