Chapter 5: Linear Equations and Inequalities

Solve · page 96: 1. Maximize f (x, y) = 2x + 5y; subject to the constraints 2y—-x<8 ; x-y<4 ; x>0; y=0

Solution $$\text{Associated equation 1: } 2y - x = 8$$ $$\text{Associated equation 2: } x - y = 4$$ $$\text{For } 2y - x = 8 \text{, put } x = 0 \implies 2y = 8 \implies y = 4 \text{ (point } (0, 4) \text{)}$$ $$\text{For } 2y - x = 8 \text{, put } y = 0 \implies -x = 8 \implies x = -8 \text{ (point } (-8, 0) \text{)}$$ $$\text{For } x - y = 4 \text{, put } x = 0 \implies -y = 4 \implies y = -4 \text{ (point } (0, -4) \text{)}$$ $$\text{For } x - y = 4 \text{, put } y = 0 \implies x = 4 \text{ (point } (4, 0) \text{)}$$ $$\text{Test } (0, 0) \text{ in } 2y - x \le 8 \implies 2(0) - 0 \le 8 \implies 0 \le 8 \text{ (True)}$$ $$\text{Test } (0, 0) \text{ in } x - y \le 4 \implies 0 - 0 \le 4 \implies 0 \le 4 \text{ (True)}$$ $$\text{Intersection of } 2y - x = 8 \text{ and } x - y = 4 \implies x = y + 4$$ $$2y - (y + 4) = 8 \implies y - 4 = 8 \implies y = 12$$ $$x = 12 + 4 = 16 \text{ (Intersection point } (16, 12) \text{)}$$ $$\text{Vertices of the feasible region: } (0, 0), (4, 0), (0, 4), \text{ and } (16, 12)$$ $$f(0, 0) = 2(0) + 5(0) = 0$$ $$f(4, 0) = 2(4) + 5(0) = 8$$ $$f(0, 4) = 2(0) + 5(4) = 20$$ $$f(16, 12) = 2(16) + 5(12) = 32 + 60 = 92$$ The maximum value of the objective function is $92$ at the point $(16, 12)$.

Explanation

• Converted the first inequality constraint into its corresponding linear equation.
• Converted the second inequality constraint into its corresponding linear equation.
• Found the intercepts of the first line on the coordinate axes by setting each variable to zero.
• Found the intercepts of the second line on the coordinate axes by setting each variable to zero.
• Tested the origin $(0, 0)$ in the first inequality to determine the shaded region containing the solution.
• Tested the origin $(0, 0)$ in the second inequality to determine its shaded region.
• Solved the system of simultaneous linear equations to find the intersection point of the boundary lines.
• Identified all the corner points (vertices) of the feasible region bounded by the constraints and non-negativity conditions.
• Evaluated the objective function $f(x, y) = 2x + 5y$ at the first vertex $(0, 0)$.
• Evaluated the objective function at the second vertex $(4, 0)$.
• Evaluated the objective function at the third vertex $(0, 4)$.
• Evaluated the objective function at the fourth vertex $(16, 12)$.
• Compared the evaluated values to find the greatest result, identifying the maximum value of the objective function.