Finding The Minimum Value Of C = 10x + 3y A Step By Step Guide

Hey guys! Ever stumbled upon a problem that looks like a bunch of lines and inequalities and wondered how to find the minimum or maximum value of something? Well, you're in the right place! Today, we're diving deep into a classic optimization problem. We'll be tackling how to find the minimum value of ${ C = 10x + 3y }$ given a set of constraints. Trust me, it’s not as scary as it sounds. We'll break it down step-by-step, making it super easy to understand. So, buckle up, grab your favorite beverage, and let's get started!

Understanding the Problem: Linear Programming

At its core, this problem is a linear programming challenge. Linear programming is a mathematical technique used to find the best possible outcome (either maximum profit or minimum cost) from a set of linear relationships. Think of it like this: you've got a bunch of rules (constraints) and you want to find the sweet spot that gives you the best result for a specific goal (objective function). The objective function here is ${ C = 10x + 3y }$, and we're trying to minimize it. The constraints are the inequalities that define the feasible region – the area where our solutions can live. These constraints are:

• ${ -x + y \geq 0 }$
• ${ 2x + y \geq 4 }$
• ${ 2x + y \leq 13 }$
• ${ x \geq 0 }$
• ${ y \geq 0 }$

These inequalities create a polygon, and the minimum value of ${ C }$ will occur at one of the vertices (corners) of this polygon. This is a fundamental principle in linear programming. Why? Because the function ${ C = 10x + 3y }$ represents a line, and as we move this line across the feasible region, the minimum and maximum values will always be at the corners. So, our mission is to find these corners and test them out! First, we'll convert these inequalities into equations to find the points of intersection, which are our vertices. This involves a bit of algebra, but nothing we can't handle! Remember, each line represents a boundary, and the points where these boundaries intersect are the key to unlocking our solution. We will graph these lines to visualize the feasible region, making it easier to identify the vertices. Graphing helps us see the problem geometrically, and it's a crucial step in understanding the solution. The feasible region is the area where all the inequalities are satisfied simultaneously. It's like a common ground where all the conditions are met. And the vertices? They're the extreme points of this common ground, the places where the lines intersect and change direction.

Step 1: Graphing the Constraints

To visualize the feasible region, we need to graph each inequality. Let’s start by converting each inequality into an equation. This will give us the lines that define the boundaries of our region. Remember, a line divides the plane into two halves, and our inequalities will tell us which half to consider. The graphical method is a fantastic way to solve linear programming problems because it provides a visual representation of the solution space. It helps us understand the constraints and how they interact with each other. Here's how we'll transform each inequality into an equation:

1. ${ -x + y \geq 0 }$ becomes ${ y = x }$
2. ${ 2x + y \geq 4 }$ becomes ${ y = -2x + 4 }$
3. ${ 2x + y \leq 13 }$ becomes ${ y = -2x + 13 }$
4. ${ x \geq 0 }$ is the y-axis
5. ${ y \geq 0 }$ is the x-axis

Now, let's graph these lines. You can use graph paper, a graphing calculator, or even online tools like Desmos. Plot each line and then determine which side of the line satisfies the original inequality. For example, for ${ -x + y \geq 0 }$, you can test a point (like (1, 1)) to see if it satisfies the inequality. If it does, the region containing that point is the solution region for that inequality. Similarly, we will test points for the other inequalities to find their respective solution regions. The region where all the solution regions overlap is our feasible region. This region is a polygon, and its vertices are the points we need to find. These points are where the lines intersect, and they represent potential solutions to our problem. To find these points, we'll solve pairs of equations simultaneously. This is where our algebra skills come in handy! We'll use methods like substitution or elimination to find the x and y coordinates of the intersection points. Once we have these coordinates, we'll plug them into our objective function to see which one gives us the minimum value.

Step 2: Finding the Vertices of the Feasible Region

The vertices are the corner points of the feasible region. These are the points where two or more constraint lines intersect. To find these points, we need to solve pairs of equations simultaneously. Let's break it down:

1. Intersection of ${ y = x }$ and ${ 2x + y = 4 }$:

   • Substitute ${ y = x }$ into ${ 2x + y = 4 }$:
   • ${ 2x + x = 4 }$
   • ${ 3x = 4 }$
   • ${ x = \frac{4}{3} }$
   • Since ${ y = x }$, then ${ y = \frac{4}{3} }$
   • So, the intersection point is ${ (\frac{4}{3}, \frac{4}{3}) }$

2. Intersection of ${ y = x }$ and ${ 2x + y = 13 }$:

   • Substitute ${ y = x }$ into ${ 2x + y = 13 }$:
   • ${ 2x + x = 13 }$
   • ${ 3x = 13 }$
   • ${ x = \frac{13}{3} }$
   • Since ${ y = x }$, then ${ y = \frac{13}{3} }$
   • So, the intersection point is ${ (\frac{13}{3}, \frac{13}{3}) }$

3. Intersection of ${ 2x + y = 4 }$ and ${ y = 0 }$:

   • Substitute ${ y = 0 }$ into ${ 2x + y = 4 }$:
   • ${ 2x + 0 = 4 }$
   • ${ 2x = 4 }$
   • ${ x = 2 }$
   • So, the intersection point is ${ (2, 0) }$

4. Intersection of ${ 2x + y = 13 }$ and ${ x = 0 }$:

   • Substitute ${ x = 0 }$ into ${ 2x + y = 13 }$:
   • ${ 2(0) + y = 13 }$
   • ${ y = 13 }$
   • So, the intersection point is ${ (0, 13) }$

5. Intersection of ${ y = x }$ and ${ x = 0 }$:

   • Since ${ x = 0 }$ and ${ y = x }$, then ${ y = 0 }$
   • So, the intersection point is ${ (0, 0) }$

However, the point ${ (0, 0) }$ does not satisfy the constraint ${ 2x + y \geq 4 }$, so it's not a vertex of our feasible region. Also, the intersection of ${ 2x + y = 13 }$ and ${ y = 0 }$ will give us a point outside the feasible region. Thus, we have four vertices: ${ (\frac{4}{3}, \frac{4}{3}), (\frac{13}{3}, \frac{13}{3}), (2, 0), }$ and ${ (0, 13) }$. Now that we have our vertices, the next step is to plug these points into our objective function. This will tell us the value of ${ C }$ at each corner of our feasible region. The smallest value we get will be the minimum value of ${ C }$, which is what we're looking for. This process is straightforward but crucial, as it's the final step in finding our solution. We're essentially evaluating our objective function at each possible extreme point to see which one gives us the best outcome.

Step 3: Evaluating the Objective Function

Now that we have the vertices, we'll plug each one into our objective function ${ C = 10x + 3y }$ to see which one gives us the minimum value.

1. For the vertex ${ (\frac{4}{3}, \frac{4}{3}) }$:

   • ${ C = 10(\frac{4}{3}) + 3(\frac{4}{3}) }$
   • ${ C = \frac{40}{3} + \frac{12}{3} }$
   • ${ C = \frac{52}{3} \approx 17.33 }$

2. For the vertex ${ (\frac{13}{3}, \frac{13}{3}) }$:

   • ${ C = 10(\frac{13}{3}) + 3(\frac{13}{3}) }$
   • ${ C = \frac{130}{3} + \frac{39}{3} }$
   • ${ C = \frac{169}{3} \approx 56.33 }$

3. For the vertex ${ (2, 0) }$:

   • ${ C = 10(2) + 3(0) }$
   • ${ C = 20 }$

4. For the vertex ${ (0, 13) }$:

   • ${ C = 10(0) + 3(13) }$
   • ${ C = 39 }$

Comparing these values, we see that the smallest value of ${ C }$ is ${ \frac{52}{3} }$ (approximately 17.33), which occurs at the vertex ${ (\frac{4}{3}, \frac{4}{3}) }$. This means that the minimum value of our objective function, subject to the given constraints, is approximately 17.33. We've successfully found the sweet spot! This whole process highlights the power of linear programming in solving optimization problems. By understanding the constraints and the objective function, we can systematically find the best possible solution. It's like navigating a maze – the constraints are the walls, and the objective function guides us to the exit in the most efficient way. So, guys, the minimum value of ${ C }$ is ${ \frac{52}{3} }$, and we found it by graphing the constraints, identifying the vertices of the feasible region, and evaluating the objective function at those vertices. That's how we crack these types of problems!

Conclusion

So, there you have it! Finding the minimum value of ${ C = 10x + 3y }$ subject to the given constraints isn't as daunting as it first appears. By understanding the principles of linear programming, graphing the constraints, finding the vertices of the feasible region, and evaluating the objective function, we can systematically solve these problems. Remember, the minimum (or maximum) value will always occur at a vertex. This approach is not just for math problems; it's used in various real-world applications, from business and economics to engineering and logistics. Understanding how to optimize outcomes is a valuable skill, and linear programming provides a powerful framework for doing so. Whether you're minimizing costs, maximizing profits, or allocating resources efficiently, the techniques we've discussed today can help you make informed decisions. So, next time you encounter an optimization problem, remember the steps we've covered, and you'll be well-equipped to find the best solution. Keep practicing, keep exploring, and keep optimizing! You've got this! And if you ever get stuck, remember, there are plenty of resources available online and in textbooks to help you along the way. The world of optimization is vast and exciting, and there's always something new to learn. So, embrace the challenge, and enjoy the journey!