Let's solve the problem step by step.
1. Understand and Define Variables:
- Given a two-digit number, let [tex]\( x \)[/tex] be the tens digit and [tex]\( y \)[/tex] be the ones digit.
- The original number can be represented as [tex]\( 10x + y \)[/tex].
- The number formed by reversing the digits is [tex]\( 10y + x \)[/tex].
2. Formulate Equations Based on the Problem Statement:
- From the problem, we know the sum of the tens and ones digits is 7. Therefore:
[tex]\[
x + y = 7 \quad \text{(Equation 1)}
\][/tex]
- The problem also states that when the digits are reversed, the new number is 2 greater than twice the original number. This can be written as:
[tex]\[
10y + x = 2 \cdot (10x + y) + 2 \quad \text{(Equation 2)}
\][/tex]
3. Simplify Equation 2:
- Start by expanding the right side of Equation 2:
[tex]\[
10y + x = 20x + 2y + 2
\][/tex]
- Rearrange this equation to isolate terms involving [tex]\( x \)[/tex] and [tex]\( y \)[/tex]:
[tex]\[
10y + x - 20x - 2y = 2
\][/tex]
[tex]\[
8y - 19x = 2 \quad \text{(Equation 3)}
\][/tex]
4. Solve the System of Equations:
- We now solve the system of equations [tex]\( x + y = 7 \)[/tex] (Equation 1) and [tex]\( 8y - 19x = 2 \)[/tex] (Equation 3).
- From Equation 1:
[tex]\[
y = 7 - x
\][/tex]
- Substitute [tex]\( y \)[/tex] from Equation 1 into Equation 3:
[tex]\[
8(7 - x) - 19x = 2
\][/tex]
[tex]\[
56 - 8x - 19x = 2
\][/tex]
[tex]\[
56 - 27x = 2
\][/tex]
[tex]\[
27x = 54
\][/tex]
[tex]\[
x = 2
\][/tex]
- Substitute [tex]\( x = 2 \)[/tex] back into Equation 1 to find [tex]\( y \)[/tex]:
[tex]\[
y = 7 - 2 = 5
\][/tex]
5. Determine the Original Number:
- The tens digit is [tex]\( x = 2 \)[/tex] and the ones digit is [tex]\( y = 5 \)[/tex].
- The original number is:
[tex]\[
10x + y = 10 \cdot 2 + 5 = 25
\][/tex]
Thus, the original number is [tex]\( \boxed{25} \)[/tex].
