The sum of a two-digit number and the number obtained by reversing the digits is 66. If the digits of the number differ by 2, find the number. How many such numbers are there?

Step 1: Represent the two-digit number

We can represent a two-digit number using its tens digit, $x$, and its units digit, $y$. So, the number can be written as $10x + y$ (e.g., if $x=5$ and $y=6$, the number is $10(5)+6=56$). The problem refers to this representation in <ref_1>.

Step 2: Represent the number with reversed digits

When the digits are reversed, the units digit becomes the tens digit and the tens digit becomes the units digit. So, $y$ becomes the tens digit and $x$ becomes the units digit, forming the number $10y + x$.

Step 3: Form the first equation based on the sum

The problem states that the sum of the original two-digit number and the number obtained by reversing its digits is 66. We set up an equation reflecting this condition.

Step 4: Simplify the first equation

Combine like terms in the sum equation. We have $10x+x = 11x$ and $y+10y = 11y$. Then, divide the entire equation by 11 to simplify it, resulting in Equation 1 as $x + y = 6$.

Step 5: Form the second equation based on digit difference

The problem states that the digits differ by 2. This means there are two possibilities: either $x$ is greater than $y$ ($x - y = 2$), or $y$ is greater than $x$ ($y - x = 2$). We will solve for both cases as stated in the example in <ref_1>.

Step 6: Solve Case 1: $x - y = 2$

We now have a system of two linear equations: $x + y = 6$ and $x - y = 2$. We will solve this system using the elimination method, as demonstrated in example 9 of <ref_2>.

Step 7: Add the equations for Case 1

Adding Equation 1 and Equation 2a eliminates the $y$ variable, allowing us to solve for $x$. We find that $2x = 8$, which simplifies to $x=4$.

Step 8: Substitute $x$ to find $y$ for Case 1

Substitute the value of $x=4$ into Equation 1 ($x + y = 6$) to find the value of $y$. This gives us $4+y=6$, which results in $y=2$.

Step 9: Determine the number for Case 1

With $x=4$ and $y=2$, the original two-digit number is $10(4) + 2 = 42$. We can verify this: $42 + 24 = 66$, and $4-2=2$. This matches the conditions.

Step 10: Solve Case 2: $y - x = 2$

For the second case, we consider $y - x = 2$, which can be rewritten as $-x + y = 2$. We again use the elimination method with Equation 1.

Step 11: Add the equations for Case 2

Adding Equation 1 and Equation 2b eliminates the $x$ variable, allowing us to solve for $y$. We find that $2y = 8$, which simplifies to $y=4$.

Step 12: Substitute $y$ to find $x$ for Case 2

Substitute the value of $y=4$ into Equation 1 ($x + y = 6$) to find the value of $x$. This gives us $x+4=6$, which results in $x=2$.

Step 13: Determine the number for Case 2

With $x=2$ and $y=4$, the original two-digit number is $10(2) + 4 = 24$. We can verify this: $24 + 42 = 66$, and $4-2=2$. This also matches the conditions.

Step 14: Count the number of solutions

We found two possible numbers that satisfy both given conditions. These numbers are 42 and 24.

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