If `5x^(2) -4xy +y^(2) -2x +1=0` then value of `x+y` is

Options

A. 1
B. 0
C. `-3`
D. 3

Answer

To solve the equation \(5x^2 - 4xy + y^2 - 2x + 1 = 0\) and find the value of \(x + y\), we can follow these steps:

### Step 1: Rearrange the equation
We start with the equation:
\[ 5x^2 - 4xy + y^2 - 2x + 1 = 0 \]

### Step 2: Group terms
We can rearrange the terms in the equation to identify perfect squares:
\[ 5x^2 - 4xy + y^2 + (-2x + 1) = 0 \]

### Step 3: Identify perfect squares
Notice that \(5x^2 - 4xy + y^2\) can be rewritten as:
\[ (2x - y)^2 \]
Thus, we can express the equation as:
\[ (2x - y)^2 + (-2x + 1) = 0 \]

### Step 4: Set each part to zero
For the equation to hold true, both parts must equal zero:
1. \( (2x - y)^2 = 0 \)
2. \( -2x + 1 = 0 \)

### Step 5: Solve the first equation
From \( (2x - y)^2 = 0 \), we get:
\[ 2x - y = 0 \implies y = 2x \]

### Step 6: Solve the second equation
From \( -2x + 1 = 0 \), we find:
\[ 2x = 1 \implies x = \frac{1}{2} \]

### Step 7: Substitute to find \(y\)
Now substitute \(x = \frac{1}{2}\) into \(y = 2x\):
\[ y = 2 \times \frac{1}{2} = 1 \]

### Step 8: Calculate \(x + y\)
Now we can find \(x + y\):
\[ x + y = \frac{1}{2} + 1 = \frac{3}{2} \]

### Final Answer
Thus, the value of \(x + y\) is:
\[ \boxed{\frac{3}{2}} \]