A two-digit number is such that the ten's digit exceeds twice the unit's digit by 2 and the number obtained by inter-changing the digits is 5 more than three times the sum of the digits. Find the two digit number.

Answer

To solve the problem, we need to set up a system of equations based on the information given in the question. Let's break it down step by step.

### Step 1: Define the Variables
Let:
- \( x \) = the ten's digit of the two-digit number
- \( y \) = the unit's digit of the two-digit number

### Step 2: Formulate the First Equation
According to the problem, the ten's digit exceeds twice the unit's digit by 2. This can be expressed as:
\[ x = 2y + 2 \]
This is our **Equation 1**.

### Step 3: Formulate the Second Equation
The problem also states that the number obtained by interchanging the digits is 5 more than three times the sum of the digits. The number with the digits interchanged is \( 10y + x \). The sum of the digits is \( x + y \). Thus, we can write:
\[ 10y + x = 3(x + y) + 5 \]
Expanding this gives:
\[ 10y + x = 3x + 3y + 5 \]
Rearranging this equation leads to:
\[ 10y + x - 3x - 3y = 5 \]
Combining like terms results in:
\[ 7y - 2x = 5 \]
This is our **Equation 2**.

### Step 4: Substitute Equation 1 into Equation 2
Now we can substitute the expression for \( x \) from Equation 1 into Equation 2:
\[ 7y - 2(2y + 2) = 5 \]
Expanding this gives:
\[ 7y - 4y - 4 = 5 \]
Combining like terms results in:
\[ 3y - 4 = 5 \]

### Step 5: Solve for \( y \)
Adding 4 to both sides:
\[ 3y = 9 \]
Dividing by 3 gives:
\[ y = 3 \]

### Step 6: Solve for \( x \)
Now that we have \( y \), we can find \( x \) using Equation 1:
\[ x = 2y + 2 = 2(3) + 2 = 6 + 2 = 8 \]

### Step 7: Form the Two-Digit Number
The two-digit number can be formed as:
\[ 10x + y = 10(8) + 3 = 80 + 3 = 83 \]

### Final Answer
Thus, the two-digit number is **83**.

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