Alright, let's tackle the problem step-by-step.
### Problem Statement:
A two-digit number has the sum of its ones and tens digit equal to ten. If the digits of the number are reversed, it forms a new number that exceeds the original number by 54. We need to find this two-digit number.
### Step-by-Step Solution:
1. Set Up Variables:
- Let [tex]\( x \)[/tex] be the tens digit of the original number.
- Let [tex]\( y \)[/tex] be the ones digit of the original number.
2. Formulate Equations:
- According to the problem, the sum of the tens and ones digit is 10:
[tex]\[ x + y = 10 \][/tex]
- When the digits are reversed, the new number formed exceeds the original number by 54. The original number is [tex]\( 10x + y \)[/tex] and the new number is [tex]\( 10y + x \)[/tex]. This gives us:
[tex]\[ (10y + x) = (10x + y) + 54 \][/tex]
3. Simplify the Second Equation:
- Rearrange the second equation:
[tex]\[ 10y + x - y - 10x = 54 \][/tex]
- Combine like terms:
[tex]\[ 9y - 9x = 54 \][/tex]
- Simplify by dividing both sides by 9:
[tex]\[ y - x = 6 \][/tex]
4. Solve the System of Equations:
- Now we have two equations:
[tex]\[ 1) \quad x + y = 10 \][/tex]
[tex]\[ 2) \quad y - x = 6 \][/tex]
To solve these equations, we can add them together to eliminate [tex]\( x \)[/tex]:
[tex]\[ (x + y) + (y - x) = 10 + 6 \][/tex]
[tex]\[ x + y + y - x = 16 \][/tex]
[tex]\[ 2y = 16 \][/tex]
[tex]\[ y = 8 \][/tex]
- Substitute [tex]\( y = 8 \)[/tex] back into the first equation:
[tex]\[ x + 8 = 10 \][/tex]
[tex]\[ x = 2 \][/tex]
5. Form the Original Number:
- The original number is formed as [tex]\( 10x + y \)[/tex]:
[tex]\[ 10 \cdot 2 + 8 = 28 \][/tex]
### Conclusion:
The original two-digit number is:
[tex]\[ \boxed{28} \][/tex]
