Let `veca =-2 hati + hatj, vecb= hati + 2hatjand vec c = 4 hati+3 hatj.` Find the value of x and y such that `vecc = x veca + y vecb.`

Answer

To solve the problem, we need to express vector \( \vec{c} \) in terms of vectors \( \vec{a} \) and \( \vec{b} \) and find the values of \( x \) and \( y \).

### Step 1: Write down the vectors
We have:
- \( \vec{a} = -2 \hat{i} + \hat{j} \)
- \( \vec{b} = \hat{i} + 2 \hat{j} \)
- \( \vec{c} = 4 \hat{i} + 3 \hat{j} \)

### Step 2: Set up the equation
According to the problem, we need to find \( x \) and \( y \) such that:
\[ \vec{c} = x \vec{a} + y \vec{b} \]
Substituting the values of \( \vec{a} \), \( \vec{b} \), and \( \vec{c} \):
\[ 4 \hat{i} + 3 \hat{j} = x(-2 \hat{i} + \hat{j}) + y(\hat{i} + 2 \hat{j}) \]

### Step 3: Expand the right-hand side
Expanding the right-hand side:
\[ 4 \hat{i} + 3 \hat{j} = (-2x + y) \hat{i} + (x + 2y) \hat{j} \]

### Step 4: Set up the system of equations
From the equation above, we can equate the coefficients of \( \hat{i} \) and \( \hat{j} \):
1. For \( \hat{i} \):
\[ -2x + y = 4 \quad \text{(1)} \]
2. For \( \hat{j} \):
\[ x + 2y = 3 \quad \text{(2)} \]

### Step 5: Solve the system of equations
We have the following system of equations:
1. \( -2x + y = 4 \)
2. \( x + 2y = 3 \)

Let's solve these equations.

From equation (1):
\[ y = 4 + 2x \quad \text{(3)} \]

Substituting (3) into equation (2):
\[ x + 2(4 + 2x) = 3 \]
\[ x + 8 + 4x = 3 \]
\[ 5x + 8 = 3 \]
\[ 5x = 3 - 8 \]
\[ 5x = -5 \]
\[ x = -1 \]

Now substituting \( x = -1 \) back into equation (3):
\[ y = 4 + 2(-1) \]
\[ y = 4 - 2 \]
\[ y = 2 \]

### Step 6: Final answer
Thus, the values of \( x \) and \( y \) are:
\[ x = -1, \quad y = 2 \]