Maximise (Z=$4x_1+5x_2$) subject to $x_1+2x_2\le 8$ $3x_1+x_2\le 9$ $x_1,x_2\ge 0$. Solve this linear programme using the simplex method and give the optimal values of ($x_1$,$x_2$) and (Z).

Step-by-step solution

1. Convert inequalities to equalities by adding slack variables ($s_1$,$s_2$): $x_1+2x_2+s_1=8$ $3x_1+x_2+s_2=9$.
2. Write the initial simplex tableau with basic variables ($s_1$,$s_2$).
3. Identify the entering variable with the most negative coefficient in the objective row (here ($x_2$) or ($x_1$) depending on tableau).
4. Perform the minimum ratio test to choose the leaving variable (compute (8/(2)) and (9/(1)), choose ($s_1$)).
5. Pivot on the chosen element to update the tableau and repeat until all reduced costs are non‑negative.
6. Obtain the final tableau where basic variables are ($x_1$=2), ($x_2$=3).
7. Read the optimal objective value (Z=4(2)+5(3)=23).

Concept tested: Simplex algorithm

Published 4 October 2026 · How this solution was prepared and checked