Step-by-step Solution:

To find the values of \( m \) and \( n \) such that \( \vec{c} = m\vec{a} + n\vec{b} \), we can proceed with the following steps: 1. Given Vectors: \[ \vec{a} = -4\hat{i} + 2\hat{j} \] \[ \vec{b} = 2\hat{i} + \hat{j} \] \[ \vec{c} = 2\hat{i} + 3\hat{j} \] 2. Express \( \vec{c} \) in Terms of \( \vec{a} \) and \( \vec{b} \): \[ \vec{c} = m\vec{a} + n\vec{b} \] Substitute the given vectors: \[ 2\hat{i} + 3\hat{j} = m(-4\hat{i} + 2\hat{j}) + n(2\hat{i} + \hat{j}) \] \[ 2\hat{i} + 3\hat{j} = (-4m + 2n)\hat{i} + (2m + n)\hat{j} \] 3. Set Up Equations by Comparing Components: - For \( \hat{i} \) components: \[ -4m + 2n = 2 \] - For \( \hat{j} \) components: \[ 2m + n = 3 \] 4. Solve the System of Equations: From the second equation: \[ n = 3 - 2m \] Substitute \( n = 3 - 2m \) into the first equation: \[ -4m + 2(3 - 2m) = 2 \] \[ -4m + 6 - 4m = 2 \] \[ -8m + 6 = 2 \] \[ -8m = -4 \] \[ m = \frac{1}{2} \] Substitute \( m = \frac{1}{2} \) back into \( n = 3 - 2m \): \[ n = 3 - 2\left(\frac{1}{2}\right) = 3 - 1 = 2 \] 5. Find \( m + n \): \[ m + n = \frac{1}{2} + 2 = \frac{5}{2} \] Therefore, the value of \( m + n \) is: \[ \boxed{\frac{5}{2}} \]