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 step by step, we need to find the values of \( x \) and \( y \) such that 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 given 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 vectors into the equation
Substituting \( \vec{u} \), \( \vec{v} \), and \( \vec{w} \) into the equation \( 2 \vec{u} - \vec{v} = \vec{w} \):
\[ 2(x \vec{a} + 2y \vec{b}) - (-2y \vec{a} + 3x \vec{b}) = 4 \vec{a} - 2 \vec{b} \]

### Step 3: Simplify the left-hand side
Calculating the left-hand side:
\[ 2(x \vec{a} + 2y \vec{b}) + 2y \vec{a} - 3x \vec{b} = 2x \vec{a} + 4y \vec{b} + 2y \vec{a} - 3x \vec{b} \]
Combine like terms:
\[ (2x + 2y) \vec{a} + (4y - 3x) \vec{b} \]

### Step 4: Set the left-hand side equal to the right-hand side
Now, we set the simplified left-hand side equal to the right-hand side:
\[ (2x + 2y) \vec{a} + (4y - 3x) \vec{b} = 4 \vec{a} - 2 \vec{b} \]

### Step 5: Equate coefficients of \( \vec{a} \) and \( \vec{b} \)
From the equation, we can equate the coefficients of \( \vec{a} \) and \( \vec{b} \):
1. For \( \vec{a} \): \( 2x + 2y = 4 \)
2. For \( \vec{b} \): \( 4y - 3x = -2 \)

### Step 6: Solve the first equation for \( y \)
From the first equation:
\[ 2x + 2y = 4 \implies x + y = 2 \implies y = 2 - x \]

### Step 7: Substitute \( y \) into the second equation
Now substitute \( y \) into the second equation:
\[ 4(2 - x) - 3x = -2 \]
Expanding this gives:
\[ 8 - 4x - 3x = -2 \implies 8 - 7x = -2 \]

### Step 8: Solve for \( x \)
Rearranging gives:
\[ -7x = -2 - 8 \implies -7x = -10 \implies x = \frac{10}{7} \]

### Step 9: Substitute \( x \) back to find \( y \)
Now substitute \( x \) back to find \( y \):
\[ y = 2 - x = 2 - \frac{10}{7} = \frac{14}{7} - \frac{10}{7} = \frac{4}{7} \]

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