If a, b, are rational numbers and `(a - 1) sqrt( 2) + 3 = b sqrt(2) + a` , then value of (a + b) is

Options

A. `-5`
B. 3
C. `-3`
D. 5

Answer

To solve the equation \((a - 1) \sqrt{2} + 3 = b \sqrt{2} + a\), we will follow these steps:

### Step 1: Rearrange the equation
Start with the given equation:
\[ (a - 1) \sqrt{2} + 3 = b \sqrt{2} + a \]
We can rearrange it to group the terms involving \(\sqrt{2}\) on one side and the constant terms on the other side:
\[ (a - 1) \sqrt{2} - b \sqrt{2} = a - 3 \]

### Step 2: Factor out \(\sqrt{2}\)
Now, factor out \(\sqrt{2}\) from the left side:
\[ ((a - 1) - b) \sqrt{2} = a - 3 \]

### Step 3: Compare coefficients
Since \(\sqrt{2}\) is an irrational number, for the equality to hold, the coefficients of \(\sqrt{2}\) on both sides must be equal, and the constant terms must also be equal. This gives us two equations:
1. \(a - 1 - b = 0\) (coefficient of \(\sqrt{2}\))
2. \(a - 3 = 0\) (constant term)

### Step 4: Solve the first equation
From the second equation \(a - 3 = 0\), we can solve for \(a\):
\[ a = 3 \]

### Step 5: Substitute \(a\) into the first equation
Now substitute \(a = 3\) into the first equation:
\[ 3 - 1 - b = 0 \]
This simplifies to:
\[ 2 - b = 0 \]
Thus, we find:
\[ b = 2 \]

### Step 6: Calculate \(a + b\)
Now that we have the values of \(a\) and \(b\):
\[ a = 3, \quad b = 2 \]
We can find \(a + b\):
\[ a + b = 3 + 2 = 5 \]

### Final Answer
The value of \(a + b\) is:
\[ \boxed{5} \]