Solution Check: Is (7, 3) A Solution?

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Is (7, 3) a Solution to the System of Inequalities?

Hey guys! Today, we are diving into a math problem where we need to figure out if a specific point, (7, 3), is a solution to a system of inequalities. Don't worry, it sounds more complicated than it actually is. We'll break it down step by step so it’s super easy to follow. Let’s get started!

The system of inequalities we're working with is:

x + 2y > 12
3x + 16y ≤ 1

Our mission is to determine whether the point (7, 3) satisfies both of these inequalities. If it does, then (7, 3) is indeed a solution to the system. If it doesn't satisfy even one of them, then it's not a solution. So, how do we do this? Let’s jump right into it.

Step-by-Step Solution

Step 1: Substitute the Point (7, 3) into the First Inequality

Our first inequality is:

x + 2y > 12

To check if (7, 3) satisfies this, we substitute x = 7 and y = 3 into the inequality:

7 + 2(3) > 12

Now, let's simplify:

7 + 6 > 12
13 > 12

So, is 13 greater than 12? Yes, it is! That means the point (7, 3) satisfies the first inequality. Great start!

Step 2: Substitute the Point (7, 3) into the Second Inequality

Now let’s move on to the second inequality:

3x + 16y ≤ 1

Again, we substitute x = 7 and y = 3 into the inequality:

3(7) + 16(3) ≤ 1

Let's simplify this as well:

21 + 48 ≤ 1
69 ≤ 1

Okay, is 69 less than or equal to 1? Nope, definitely not. 69 is way bigger than 1. This means that the point (7, 3) does not satisfy the second inequality.

Step 3: Determine if (7, 3) is a Solution to the System

Remember, for (7, 3) to be a solution to the system of inequalities, it needs to satisfy both inequalities. We found that it satisfies the first one (13 > 12), but it doesn't satisfy the second one (69 ≤ 1). Therefore, (7, 3) is not a solution to the system of inequalities.

Why This Matters: Understanding Systems of Inequalities

So, why is it important to check both inequalities? Think of it like this: a system of inequalities represents a set of conditions that all need to be true at the same time. If a point doesn't meet all the conditions, it's not a solution to the entire system. This concept is super useful in many real-world applications.

For example, in business, you might have constraints on production costs and sales prices. You need to find a solution (like the number of items to produce) that satisfies both the cost constraints and the price constraints to make a profit. In engineering, you might have constraints on the strength of materials and the weight they can support. You need to design a structure that meets both requirements to ensure safety and stability.

Understanding systems of inequalities helps us solve problems where we have multiple conditions to meet simultaneously. It’s a fundamental concept in math that has wide-ranging applications.

Common Mistakes to Avoid

When working with systems of inequalities, there are a few common mistakes that students often make. Let’s go over them so you can avoid these pitfalls:

  1. Forgetting to Check Both Inequalities: This is the most common mistake. As we saw, a point must satisfy all inequalities in the system to be a solution. Don't stop after checking just one!
  2. Incorrectly Substituting Values: Make sure you substitute the x and y values correctly into the inequalities. Double-check your work to avoid simple errors.
  3. Making Arithmetic Errors: Math can be tricky, and it’s easy to make a small arithmetic mistake, especially when dealing with larger numbers. Take your time and double-check your calculations.
  4. Misinterpreting the Inequality Signs: Pay close attention to the inequality signs (> , <, ≥, ≤). A simple mistake here can completely change the outcome.

By being aware of these common mistakes, you can improve your accuracy and confidence in solving systems of inequalities.

Real-World Applications of Systems of Inequalities

Okay, so we’ve solved the problem and talked about avoiding mistakes. But where does this stuff actually get used in the real world? Systems of inequalities pop up in all sorts of places. Let’s look at a few examples:

  1. Budgeting and Finance: Imagine you're trying to create a budget. You have a certain amount of money to spend, and you need to allocate it to different categories like housing, food, and entertainment. Each category has its own constraints (e.g., you can’t spend more than a certain amount on rent, and you want to save at least a certain amount each month). Systems of inequalities can help you find a spending plan that meets all your constraints.
  2. Nutrition and Diet Planning: Dietitians use systems of inequalities to plan balanced diets. You might have constraints on the number of calories, the amount of protein, and the intake of vitamins. The goal is to create a meal plan that satisfies all these nutritional requirements.
  3. Manufacturing and Production: In manufacturing, companies often have constraints on resources like labor, materials, and time. They need to determine how much of each product to produce to maximize profit while staying within these constraints. This is a classic application of linear programming, which uses systems of inequalities.
  4. Logistics and Transportation: Companies that transport goods need to optimize routes and schedules. They might have constraints on the distance traveled, the delivery time, and the capacity of vehicles. Systems of inequalities help them find the most efficient way to transport goods.
  5. Environmental Science: Environmental scientists use systems of inequalities to model and manage resources. For example, they might have constraints on the amount of pollution that can be released into the air or water, and they need to find ways to meet energy needs while staying within these environmental limits.

These are just a few examples, but they show how versatile systems of inequalities can be. By understanding how to solve them, you're gaining a valuable skill that can be applied in many different fields.

Practice Problems

Alright, let’s put what we’ve learned into practice. Here are a couple of problems for you to try on your own. Remember to follow the steps we discussed:

Practice Problem 1

Is the point (2, 4) a solution to the following system of inequalities?

2x + y > 7
x - 3y ≤ -10

Practice Problem 2

Is the point (-1, 1) a solution to the following system of inequalities?

-x + 2y ≥ 3
4x + y < 0

Go ahead and work through these problems. Check your answers by substituting the point into each inequality and seeing if it satisfies both. If you get stuck, review the steps we covered earlier in this article. Practice makes perfect, so the more you work with these types of problems, the easier they will become.

Conclusion: Mastering Systems of Inequalities

So, to wrap things up, we tackled the question: "Is (7, 3) a solution to this system of inequalities?" We walked through the steps, substituted the point into each inequality, and determined that (7, 3) is not a solution because it doesn't satisfy both inequalities.

We also discussed why understanding systems of inequalities is important and how they apply to real-world situations, from budgeting to manufacturing. We covered common mistakes to avoid and worked through some practice problems to help you solidify your understanding.

Remember, solving systems of inequalities is all about checking if a point satisfies all the conditions. If it doesn't, then it's not a solution to the entire system.

Keep practicing, and you'll become a pro at solving these types of problems in no time. Math can be challenging, but with a step-by-step approach and a bit of practice, you can conquer any problem that comes your way. Keep up the great work, guys!