(a) Cramer's rule: $x=-1,\ y=3,\ z=2$. (b) Newton forward interpolation at $u=0.5$: about $21.69$ lakhs.

(a) Solve by Cramer's rule: $x+y+z=4,\ 2x-y+3z=1,\ 3x+2y-z=1$.

$\Delta=\begin{vmatrix}1&1&1\\2&-1&3\\3&2&-1\end{vmatrix}=1(1-6)-1(-2-9)+1(4+3)=-5+11+7=13.$

$\Delta_x=\begin{vmatrix}4&1&1\\1&-1&3\\1&2&-1\end{vmatrix}=4(1-6)-1(-1-3)+1(2+1)=-20+4+3=-13,$

$\Delta_y=\begin{vmatrix}1&4&1\\2&1&3\\3&1&-1\end{vmatrix}=1(-1-3)-4(-2-9)+1(2-3)=-4+44-1=39,$

$\Delta_z=\begin{vmatrix}1&1&4\\2&-1&1\\3&2&1\end{vmatrix}=1(-1-2)-1(2-3)+4(4+3)=-3+1+28=26.$

$x=\frac{\Delta_x}{\Delta}=\frac{-13}{13}=-1,\quad y=\frac{\Delta_y}{\Delta}=\frac{39}{13}=3,\quad z=\frac{\Delta_z}{\Delta}=\frac{26}{13}=2.$

(Check: $-1+3+2=4$ ✓.)

(b) Interpolate the population for 1946. Since $1946$ is near the start of the table (base $x_0=1941,\ h=10$), use Newton's forward interpolation. Build the difference table: