Solving Systems Of Equations Finding The Value Of Y

by Kenji Nakamura 52 views

Hey guys! Let's dive into this math problem together. We've got a system of equations here, and our mission is to figure out the value of 'y'. It looks a bit like a puzzle, but don't worry, we'll break it down step by step.

Understanding the Equations

First, let's take a good look at what we're working with. We have two equations:

  1. x = 5
  2. x + y = 60

In this system of equations, we already know that x has a fixed value of 5. That's a great starting point! The second equation tells us that when we add x and y together, we should get 60. Our goal is to find the value of y that makes this true.

Keywords are essential in math problems, especially when understanding the fundamental concepts involved. Here, the keywords are "system of equations," "fixed value," and "variable." Recognizing these terms helps us understand the nature of the problem and the steps we need to take to solve it. A system of equations involves two or more equations with the same variables, and we're trying to find values for those variables that satisfy all the equations simultaneously. A fixed value, in this case, x = 5, means that the value of x is already known and won't change. A variable, like y, is a symbol that represents an unknown quantity that we need to find. Keeping these keywords in mind will help you tackle similar problems more confidently.

Solving for y: A Step-by-Step Approach

Now, let's get our hands dirty and solve for y. Since we know that x is 5, we can use this information in the second equation. This process is called substitution, a crucial technique in algebra.

Here's how it works:

  1. Start with the second equation: x + y = 60

  2. Substitute the value of x: Since x = 5, we replace x with 5 in the equation. This gives us: 5 + y = 60

  3. Isolate y: Our goal is to get y by itself on one side of the equation. To do this, we need to get rid of the 5 that's being added to y. We can do this by subtracting 5 from both sides of the equation. Remember, whatever we do to one side of an equation, we must do to the other to keep it balanced.

    So, we subtract 5 from both sides: 5 + y - 5 = 60 - 5

  4. Simplify: Now, let's simplify the equation. 5 - 5 cancels out on the left side, and 60 - 5 is 55. This leaves us with: y = 55

And there we have it! We've found the value of y. It turns out that y equals 55.

Double-Checking Our Answer

It's always a good idea to double-check our work, especially in math. To make sure our answer is correct, we can plug the values of x and y back into the original equations.

  1. First equation: x = 5 (This one is already true, as we were given this information.)
  2. Second equation: x + y = 60. Let's substitute our values: 5 + 55 = 60 60 = 60

Our values satisfy both equations! This confirms that our answer is correct. y is indeed 55.

Checking our answer is a critical step, often overlooked but incredibly important. It ensures that we haven't made any mistakes along the way. By substituting the values we found back into the original equations, we can verify that they hold true. This process not only confirms our solution but also reinforces our understanding of the problem and the relationships between the variables.

The Correct Answer: Option a) y = 55

Looking back at the choices, we can see that the correct answer is:

a) y = 55

We successfully solved the system of equations and found the value of y. Awesome job, guys!

Why Other Options Are Incorrect

Let's quickly discuss why the other options aren't correct. This will help us solidify our understanding of the problem and the solution.

  • b) y = 10: If we substitute y = 10 into the second equation, we get 5 + 10 = 15, which is not equal to 60. So, this is incorrect.
  • c) y = 5: Substituting y = 5 into the second equation gives us 5 + 5 = 10, which is also not equal to 60. This option is wrong as well.
  • d) y = 60: If we try y = 60, the second equation becomes 5 + 60 = 65, which is not 60. Therefore, this option is not correct.

Understanding why the incorrect options are wrong is just as important as knowing why the correct option is right. This deeper understanding helps prevent similar mistakes in the future and strengthens your problem-solving skills. By analyzing the errors, you can identify common pitfalls and develop strategies to avoid them.

Real-World Applications of Systems of Equations

You might be wondering, "Okay, this is cool, but when will I ever use this in real life?" Well, systems of equations pop up in all sorts of situations! They're not just abstract math problems; they have practical applications in various fields.

For instance:

  • Business: Companies use systems of equations to model costs, revenues, and profits. They can determine the break-even point (where costs equal revenue) by solving a system of equations.
  • Science: Scientists use them in chemistry to balance chemical equations, in physics to analyze forces, and in engineering to design structures.
  • Economics: Economists use systems of equations to model supply and demand, predict market behavior, and analyze economic policies.
  • Computer Graphics: In computer graphics and game development, systems of equations are used to calculate transformations, projections, and intersections.

Understanding the real-world applications of mathematical concepts can make learning them more engaging and meaningful. When you see how systems of equations are used in various fields, you realize that math isn't just about abstract formulas; it's a powerful tool for solving real-world problems. This understanding can motivate you to learn more and appreciate the relevance of mathematics in your life.

Tips for Solving Systems of Equations

Here are a few tips and tricks to keep in mind when you're tackling systems of equations:

  • Read the problem carefully: Make sure you understand what the question is asking and what information you're given.
  • Identify the variables: Determine the unknowns you need to find.
  • Choose a method: Decide whether substitution, elimination, or graphing is the best approach for the specific problem.
  • Show your work: Write down each step clearly and neatly. This will help you avoid mistakes and make it easier to check your work.
  • Check your answer: Always plug your solution back into the original equations to verify that it's correct.
  • Practice, practice, practice: The more you practice, the more comfortable you'll become with solving systems of equations.

Practice is the key to mastering any mathematical skill, including solving systems of equations. The more you work through different types of problems, the better you'll become at recognizing patterns, applying the correct techniques, and avoiding common mistakes. Don't be afraid to try different approaches and learn from your errors. With consistent practice, you'll build confidence and improve your problem-solving abilities.

Conclusion: You've Got This!

We've successfully solved the system of equations and found that y = 55. We walked through the steps, double-checked our answer, and even explored some real-world applications. You've got this, guys! Keep practicing, and you'll become a system-of-equations-solving pro in no time.

Remember, math is like a muscle; the more you use it, the stronger it gets. So, keep flexing those math muscles, and you'll be amazed at what you can achieve!