If a, b are rational numbers and `(a-1)(sqrt2)+3=bsqrt(2)+a` , then value of (a+b) is :

Options

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

Answer

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

### Step 1: Rearranging the equation
We start with the equation:
\[ (a-1)\sqrt{2} + 3 = b\sqrt{2} + a \]
We can rearrange this 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, we can factor out \(\sqrt{2}\) from the left side:
\[ ((a-1) - b)\sqrt{2} = a - 3 \]

### Step 3: Equating coefficients
Since \(\sqrt{2}\) is an irrational number, for the equation to hold true, the coefficients of \(\sqrt{2}\) on both sides must be equal, and the constant terms must also be equal. Therefore, we can set up the following equations:
1. \(a - 1 - b = 0\) (coefficient of \(\sqrt{2}\))
2. \(a - 3 = 0\) (constant terms)

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

### 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 both \(a\) and \(b\):
\[ a + b = 3 + 2 = 5 \]

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