Manual solution: From first: z2 = (-9 - 2*z3)/6 Sub into second: 2*[(-9 - 2*z3)/6] + 10*z3 = 9 → (-18 - 4*z3)/6 + 10*z3 = 9 → -3 - (2/3)z3 + 10*z3 = 9 → (28/3)z3 = 12 → z3 = 9/7 ≈ 1.285714 Then z2 = (-9 - 2*(9/7))/6 = (-9 - 18/7)/6 = (-81/7)/6 = -81/42 = -27/14 ≈ -1.92857

(x=4 to 7, h=3): a = 2 b = (5-2)/3 - 3/6*(2 1.285714 + 0) = 1 - 0.5 (2.571428) = 1 - 1.285714 = -0.285714 c = 1.285714/2 = 0.642857 d = (0 - 1.285714)/(6*3) = -1.285714/18 = -0.0714286 Step 5: Interpolate New x Values For any new x, determine the correct interval, then:

[ a = y_i ] [ b = \fracy_i+1 - y_ih_i - \frach_i6(2z_i + z_i+1) ] [ c = z_i / 2 ] [ d = \fracz_i+1 - z_i6h_i ]

(x=1 to 2, h=1): a = 2 b = (3-2)/1 - 1/6*(2 0 + (-1.92857)) = 1 - (1/6) (-1.92857) = 1 + 0.32143 = 1.32143 c = 0/2 = 0 d = (-1.92857 - 0)/(6*1) = -0.32143

Solve in Excel: Use and MMULT or manual algebra.