Step 1: Set up equations based on problem statements

Let the two-digit number be $10x + y$, where $x$ is the tens digit and $y$ is the units digit. The reversed number is $10y + x$. According to the problem, their sum is $99$: $(10x + y) + (10y + x) = 99$

$11(x + y) = 99 \implies x + y = 9$

Step 2: Use the difference between digits

We are given that the difference between the digits is $7$: $x - y = 7$

$x - y = 7$

Step 3: Solve the system of linear equations

Add the two equations $(x + y = 9)$ and $(x - y = 7)$: $(x + y) + (x - y) = 9 + 7 \implies 2x = 16 \implies x = 8$ Substituting $x = 8$ into $x + y = 9$, we get $y = 1$.

$x = 8, \quad y = 1$

Step 4: Determine the two-digit number

With $x = 8$ and $y = 1$, the two-digit number $10x + y$ is $10(8) + 1 = 81$. (If $x - y = -7$, the number would be $18$). Checking the options, $81$ is present.

$81$