`{:(" A 1"),("+1 B"),(--),(" B 0"),(--):}`

Answer

To solve the problem \( a1 + 1b = b0 \), we will break it down step by step.

### Step 1: Understand the equation
The equation \( a1 + 1b = b0 \) can be interpreted as:
- \( a1 \) represents the two-digit number where \( a \) is the tens digit and 1 is the units digit, which can be expressed as \( 10a + 1 \).
- \( 1b \) represents the two-digit number where 1 is the tens digit and \( b \) is the units digit, which can be expressed as \( 10 + b \).
- \( b0 \) represents the two-digit number where \( b \) is the tens digit and 0 is the units digit, which can be expressed as \( 10b \).

So we can rewrite the equation as:
\[ (10a + 1) + (10 + b) = 10b \]

### Step 2: Simplify the equation
Now, let's simplify the equation:
\[ 10a + 1 + 10 + b = 10b \]
Combining like terms gives:
\[ 10a + b + 11 = 10b \]

### Step 3: Rearranging the equation
Next, we will rearrange the equation to isolate \( a \) and \( b \):
\[ 10a + 11 = 10b - b \]
This simplifies to:
\[ 10a + 11 = 9b \]

### Step 4: Solve for \( b \)
From the equation \( 10a + 11 = 9b \), we can express \( b \) in terms of \( a \):
\[ b = \frac{10a + 11}{9} \]

### Step 5: Find integer values for \( a \) and \( b \)
Since \( a \) and \( b \) must be digits (0-9), we will test integer values for \( a \) from 0 to 9 to find a valid \( b \).

- If \( a = 0 \):
\[ b = \frac{10(0) + 11}{9} = \frac{11}{9} \quad \text{(not an integer)} \]
- If \( a = 1 \):
\[ b = \frac{10(1) + 11}{9} = \frac{21}{9} \quad \text{(not an integer)} \]
- If \( a = 2 \):
\[ b = \frac{10(2) + 11}{9} = \frac{31}{9} \quad \text{(not an integer)} \]
- If \( a = 3 \):
\[ b = \frac{10(3) + 11}{9} = \frac{41}{9} \quad \text{(not an integer)} \]
- If \( a = 4 \):
\[ b = \frac{10(4) + 11}{9} = \frac{51}{9} \quad \text{(not an integer)} \]
- If \( a = 5 \):
\[ b = \frac{10(5) + 11}{9} = \frac{61}{9} \quad \text{(not an integer)} \]
- If \( a = 6 \):
\[ b = \frac{10(6) + 11}{9} = \frac{71}{9} \quad \text{(not an integer)} \]
- If \( a = 7 \):
\[ b = \frac{10(7) + 11}{9} = \frac{81}{9} = 9 \quad \text{(valid)} \]
- If \( a = 8 \):
\[ b = \frac{10(8) + 11}{9} = \frac{91}{9} \quad \text{(not an integer)} \]
- If \( a = 9 \):
\[ b = \frac{10(9) + 11}{9} = \frac{101}{9} \quad \text{(not an integer)} \]

### Step 6: Conclusion
The only valid solution we found is \( a = 7 \) and \( b = 9 \).

### Final Check
To verify:
\[ a1 + 1b = 71 + 19 = 90 \]
And:
\[ b0 = 90 \]
Both sides are equal, confirming our solution is correct.

### Summary of Values
- \( a = 7 \)
- \( b = 9 \)

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