Math Problem Statement
z=x-5y的最小值为多少
Solution
To find the minimum value of under the given constraints:
4x - 3y - 3 \geq 0 \\ x - 2y - 2 \leq 0 \\ 2x + 6y - 9 \leq 0 \end{cases}$$ First, let's rewrite these inequalities for clarity: 1. $$4x - 3y \geq 3$$ 2. $$x - 2y \leq 2$$ 3. $$2x + 6y \leq 9$$ ### Step-by-Step Solution 1. **Express y in terms of x (or vice versa) from each inequality:** From the first inequality: $$4x - 3y \geq 3 \implies y \leq \frac{4x - 3}{3}$$ From the second inequality: $$x - 2y \leq 2 \implies y \geq \frac{x - 2}{2}$$ From the third inequality: $$2x + 6y \leq 9 \implies y \leq \frac{9 - 2x}{6}$$ 2. **Combine these inequalities to find the feasible region for (x, y):** We have: $$\frac{x - 2}{2} \leq y \leq \min \left( \frac{4x - 3}{3}, \frac{9 - 2x}{6} \right)$$ 3. **Graphical Method:** To find the feasible region, we need to plot these lines and determine the intersection points: - $$y = \frac{4x - 3}{3}$$ - $$y = \frac{x - 2}{2}$$ - $$y = \frac{9 - 2x}{6}$$ 4. **Finding Intersection Points:** - Intersection of $$y = \frac{4x - 3}{3}$$ and $$y = \frac{x - 2}{2}$$: $$\frac{4x - 3}{3} = \frac{x - 2}{2}$$ Solving this equation for $$x$$: $$8x - 6 = 3x - 6 \implies 5x = 0 \implies x = 0, \quad y = -1$$ - Intersection of $$y = \frac{4x - 3}{3}$$ and $$y = \frac{9 - 2x}{6}$$: $$\frac{4x - 3}{3} = \frac{9 - 2x}{6}$$ Solving this equation for $$x$$: $$8x - 6 = 9 - 2x \implies 10x = 15 \implies x = 1.5, \quad y = 1$$ - Intersection of $$y = \frac{x - 2}{2}$$ and $$y = \frac{9 - 2x}{6}$$: $$\frac{x - 2}{2} = \frac{9 - 2x}{6}$$ Solving this equation for $$x$$: $$3(x - 2) = 9 - 2x \implies 3x - 6 = 9 - 2x \implies 5x = 15 \implies x = 3, \quad y = 0.5$$ 5. **Evaluate $$z = x - 5y$$ at these intersection points:** - At $$(0, -1)$$: $$z = 0 - 5(-1) = 5$$ - At $$(1.5, 1)$$: $$z = 1.5 - 5(1) = -3.5$$ - At $$(3, 0.5)$$: $$z = 3 - 5(0.5) = 3 - 2.5 = 0.5$$ The minimum value of $$z$$ is $$-3.5$$. Thus, the minimum value of $$z = x - 5y$$ is $$\boxed{-3.5}$$. Would you like more details or have any questions? Here are some related questions you might be interested in: 1. How to graph linear inequalities? 2. How to find the intersection points of lines algebraically? 3. How to solve systems of linear inequalities? 4. How to minimize or maximize a linear function subject to constraints? 5. What are the feasible regions in linear programming? **Tip:** When solving systems of inequalities, visualizing the feasible region using a graph can greatly aid in understanding the problem and finding the solution.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Linear Programming
Systems of Inequalities
Optimization
Formulas
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Theorems
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Suitable Grade Level
Advanced High School / College
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