Given that the vectors `vec(a) and vec(b)` are non- collinear, the values of x and y for which the vector equality ` 2 vec(u) -vec(v)= vec(w)` holds true if `vec(u) = x vec(a) + 2y vec(b), vec(v)= - 2 y vec (a) + 3 x vec(b), vec(w) = 4 vec(a)-2 vec(b) ` are

Options

A. `x=(4)/(7), y= (6)/(7)`
B. `x=(10)/(7), y= (4)/(7)`
C. `x=(8)/(7), y= (2)/(7)`
D. `x=2, y= 3`

Explanation

To solve the problem, we need to find the values of \( x \) and \( y \) for which the vector equality \( 2 \vec{u} - \vec{v} = \vec{w} \) holds true, given the definitions of the vectors \( \vec{u} \), \( \vec{v} \), and \( \vec{w} \).

### Step 1: Write down the expressions for the vectors
We have:
- \( \vec{u} = x \vec{a} + 2y \vec{b} \)
- \( \vec{v} = -2y \vec{a} + 3x \vec{b} \)
- \( \vec{w} = 4 \vec{a} - 2 \vec{b} \)

### Step 2: Substitute the expressions into the vector equation
We need to substitute \( \vec{u} \) and \( \vec{v} \) into the equation \( 2 \vec{u} - \vec{v} = \vec{w} \):
\[ 2 \vec{u} = 2(x \vec{a} + 2y \vec{b}) = 2x \vec{a} + 4y \vec{b} \]
\[ -\vec{v} = -(-2y \vec{a} + 3x \vec{b}) = 2y \vec{a} - 3x \vec{b} \]
Now, combine these:
\[ 2 \vec{u} - \vec{v} = (2x \vec{a} + 4y \vec{b}) + (2y \vec{a} - 3x \vec{b}) = (2x + 2y) \vec{a} + (4y - 3x) \vec{b} \]

### Step 3: Set the expression equal to \( \vec{w} \)
Now we set this equal to \( \vec{w} \):
\[ (2x + 2y) \vec{a} + (4y - 3x) \vec{b} = (4 \vec{a} - 2 \vec{b}) \]

### Step 4: Equate coefficients
From the equation above, we can equate the coefficients of \( \vec{a} \) and \( \vec{b} \):
1. For \( \vec{a} \):
\[ 2x + 2y = 4 \quad \text{(1)} \]
2. For \( \vec{b} \):
\[ 4y - 3x = -2 \quad \text{(2)} \]

### Step 5: Solve the system of equations
From equation (1):
\[ 2x + 2y = 4 \implies x + y = 2 \implies y = 2 - x \quad \text{(3)} \]

Now substitute (3) into equation (2):
\[ 4(2 - x) - 3x = -2 \]
\[ 8 - 4x - 3x = -2 \]
\[ 8 - 7x = -2 \]
\[ -7x = -10 \implies x = \frac{10}{7} \]

Now substitute \( x \) back into equation (3) to find \( y \):
\[ y = 2 - \frac{10}{7} = \frac{14}{7} - \frac{10}{7} = \frac{4}{7} \]

### Final Result
Thus, the values of \( x \) and \( y \) are:
\[ x = \frac{10}{7}, \quad y = \frac{4}{7} \]