Maximize 5x + 7y: A Step-by-Step Guide

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Maximize 5x + 7y: A Step-by-Step Guide

Hey guys! Let's dive into a fascinating mathematical problem: maximizing the objective function 5x + 7y, subject to some constraints. This type of problem pops up all the time in fields like economics, engineering, and operations research. We're going to break it down step-by-step, so you'll not only understand the solution but also the why behind it. So, grab your thinking caps, and let's get started!

Understanding the Problem

Before we jump into calculations, let's make sure we're all on the same page. We're trying to find the highest possible value of the expression 5x + 7y. This is our objective function. However, we can't just pick any values for x and y; we have to stick to the constraints.

The constraints are inequalities that x and y must satisfy:

  1. 2x + 3y ≤ 12
  2. x + y ≤ 5
  3. x ≥ 0
  4. y ≥ 0

The first two constraints are linear inequalities, and the last two simply tell us that x and y must be non-negative. These constraints define a feasible region, which is the set of all possible (x, y) pairs that satisfy all the conditions. Our goal is to find the point within this feasible region that gives us the maximum value for 5x + 7y.

Why is this important?

You might be wondering, "Why should I care about maximizing 5x + 7y?" Well, these kinds of problems show up in all sorts of real-world situations. Imagine you're running a business, and x represents the number of one product you make, and y represents the number of another. The objective function might represent your profit, and the constraints could be limitations on resources, like labor or materials. Finding the maximum value helps you figure out the most profitable way to run your business! This is just one example, but the possibilities are endless.

Step 1: Graphing the Constraints

The best way to visualize this problem is by graphing the constraints. Each inequality represents a region in the xy-plane, and the feasible region is the area where all these regions overlap. Let's graph each constraint one by one:

Constraint 1: 2x + 3y ≤ 12

To graph this, we first treat it as an equation: 2x + 3y = 12. We can find two points on this line by setting x = 0 and solving for y, then setting y = 0 and solving for x.

  • If x = 0, then 3y = 12, so y = 4. This gives us the point (0, 4).
  • If y = 0, then 2x = 12, so x = 6. This gives us the point (6, 0).

Plot these points and draw a line through them. Now, since our original constraint is 2x + 3y ≤ 12, we need to figure out which side of the line represents the solution. We can test a point, like (0, 0). Plugging in x = 0 and y = 0, we get 2(0) + 3(0) ≤ 12, which is true. So, the region on the same side of the line as (0, 0) is the solution to this inequality. Shade this region.

Constraint 2: x + y ≤ 5

We follow the same process as above. First, we treat it as an equation: x + y = 5.

  • If x = 0, then y = 5. This gives us the point (0, 5).
  • If y = 0, then x = 5. This gives us the point (5, 0).

Plot these points and draw a line through them. Again, we test the point (0, 0): 0 + 0 ≤ 5, which is true. So, we shade the region on the same side of the line as (0, 0).

Constraints 3 and 4: x ≥ 0 and y ≥ 0

These are simple! x ≥ 0 means we're only considering the region to the right of the y-axis, and y ≥ 0 means we're only considering the region above the x-axis. In other words, we're working in the first quadrant.

The Feasible Region

Now, here's the cool part: the feasible region is the area where all the shaded regions overlap. It's a polygon formed by the intersection of all the constraints. Take a look at your graph – you should see a four-sided shape (a quadrilateral) in the first quadrant. The corners of this shape are called corner points, and they're super important, as we'll see in the next step.

Step 2: Identifying the Corner Points

The corner points are the vertices of the feasible region. These are the points where the lines representing our constraints intersect. We need to find the coordinates of these points.

From our graph, we can usually identify some of the corner points easily:

  • (0, 0): This is the origin, where the x-axis and y-axis intersect.
  • (5, 0): This is the point where the line x + y = 5 intersects the x-axis.
  • (0, 4): This is the point where the line 2x + 3y = 12 intersects the y-axis.

But what about the point where the lines 2x + 3y = 12 and x + y = 5 intersect? We can't just read it off the graph; we need to solve these equations simultaneously. Let's use the substitution method:

  1. Solve the second equation for x: x = 5 - y
  2. Substitute this into the first equation: 2(5 - y) + 3y = 12
  3. Simplify and solve for y: 10 - 2y + 3y = 12 => y = 2
  4. Substitute y = 2 back into x = 5 - y: x = 5 - 2 => x = 3

So, the fourth corner point is (3, 2).

Now we have all our corner points: (0, 0), (5, 0), (0, 4), and (3, 2). These are the key to finding the maximum value of our objective function.

Step 3: Evaluating the Objective Function at the Corner Points

Here's the magic part! A fundamental theorem in linear programming states that the maximum (or minimum) value of a linear objective function, subject to linear constraints, will always occur at one of the corner points of the feasible region. This is super helpful because it means we only need to check a few points, rather than trying every possible (x, y) combination.

Our objective function is 5x + 7y. Let's evaluate it at each corner point:

  • (0, 0): 5(0) + 7(0) = 0
  • (5, 0): 5(5) + 7(0) = 25
  • (0, 4): 5(0) + 7(4) = 28
  • (3, 2): 5(3) + 7(2) = 15 + 14 = 29

Step 4: Determining the Maximum Value

Looking at the values we calculated, the largest value of the objective function is 29, which occurs at the point (3, 2). This is our solution!

Therefore, the maximum value of 5x + 7y, subject to the given constraints, is 29, and it occurs when x = 3 and y = 2.

Wrapping Up

Guys, we did it! We successfully maximized the objective function using the graphical method. We graphed the constraints, identified the feasible region, found the corner points, and evaluated the objective function at those points. This method is a powerful tool for solving linear programming problems, and it's used in all sorts of real-world applications.

Key Takeaways

  • Objective Function: The expression you're trying to maximize or minimize.
  • Constraints: Inequalities that limit the possible values of your variables.
  • Feasible Region: The area on the graph where all constraints are satisfied.
  • Corner Points: The vertices of the feasible region. The optimal solution always occurs at a corner point.

This is just the beginning of your journey into linear programming. There are other methods to solve these problems, especially when you have more variables and constraints. But the graphical method is a great way to visualize the problem and understand the basic concepts. Keep practicing, and you'll become a pro in no time!

Hope this helps you guys understand how to maximize linear functions with constraints. Keep up the great work!