To find the value of \(k\), we need to substitute the coordinates of the point of intersection \((1, -2)\) into both equations and solve for \(k\).

Given the equations of the lines:

1. \(2y - kx + 2 = 0\)

2. \(y + x - \frac{k}{2} = 0\)

Substitute the coordinates \((1, -2)\) into the first equation:

\(2(-2) - k(1) + 2 = 0\)

\(-4 - k + 2 = 0\)

\(-k - 2 = 0\)

\(k = -2\)

Substitute the coordinates \((1, -2)\) into the second equation:

\((-2) + 1 - \frac{k}{2} = 0\)

\(-1 - \frac{k}{2} = 0\)

\(-\frac{k}{2} = 1\)

\(k = -2 \cdot 1\)

\(k = -2\)

Since both equations give us the same value of k the value of k that satisfies both equations and the point of intersection is k = -2.