Maximise (Z=3x+2y) subject to (x+2yle 10), (4x+yle 12), (x,yge 0). Solve this linear programme using the simplex method and give the optimal values of ((x,y)) and (Z).

Step-by-step solution

1. Write the constraints in equality form by adding slack variables: (x+2y+_1=10), (4x+y+_2=12), (x,y,s_1,s_2ge0).
2. Set up the initial simplex tableau with objective row (-Z+3x+2y=0).
3. Select entering variable with most negative coefficient in objective row: (x) (coefficient -3).
4. Determine leaving variable using minimum ratio test: (min{10/1,12/4}=2.5), so (s_2) leaves. Pivot on the (x) entry in row 2. After pivot, tableau becomes: Row1: (x=2), (s_1=6), (y=4), (s_2=0).
5. Check the objective row: coefficients of non‑basic variables (y) and (s_1) are non‑negative, so optimality reached.
6. Read solution from tableau: (x=2,;y=4) and (Z=3(2)+2(4)=14).

Concept tested: Simplex method for LP

Published 7 October 2026 · How this solution was prepared and checked