1. **State the problem:** Reduce the given linear programming problem to its standard form. Given: Maximize $$Z = 5x_1 + 3x_2 + 4x_2$$ Subject to: $$2x_1 - 5x_2 \leq 6$$ $$2x_1 + 3x_2 + x_3 \geq 5$$ $$3x_1 + 4x_2 \leq 3$$ with $$x_1, x_2 \geq 0$$ 2. **Simplify the objective function:** Note that $$3x_2 + 4x_2 = 7x_2$$, so the objective becomes: $$Z = 5x_1 + 7x_2$$ 3. **Convert inequalities to equalities by adding slack and surplus variables:** - For $$2x_1 - 5x_2 \leq 6$$, add slack variable $$s_1 \geq 0$$: $$2x_1 - 5x_2 + s_1 = 6$$ - For $$2x_1 + 3x_2 + x_3 \geq 5$$, subtract surplus variable $$s_2 \geq 0$$: $$2x_1 + 3x_2 + x_3 - s_2 = 5$$ - For $$3x_1 + 4x_2 \leq 3$$, add slack variable $$s_3 \geq 0$$: $$3x_1 + 4x_2 + s_3 = 3$$ 4. **List all variables and their non-negativity constraints:** $$x_1, x_2, x_3, s_1, s_2, s_3 \geq 0$$ 5. **Final standard form:** Maximize $$Z = 5x_1 + 7x_2$$ Subject to $$2x_1 - 5x_2 + s_1 = 6$$ $$2x_1 + 3x_2 + x_3 - s_2 = 5$$ $$3x_1 + 4x_2 + s_3 = 3$$ with $$x_1, x_2, x_3, s_1, s_2, s_3 \geq 0$$ This is the standard form of the given linear programming problem.