Solving Y + 3 = -y + 9 A Step-by-Step Guide

by Sam Evans 44 views
Iklan Headers

Hey guys! Today, we're diving deep into the world of algebra to tackle a common type of equation: solving for a variable. Specifically, we'll be focusing on the equation y + 3 = -y + 9. This might seem daunting at first, but trust me, with a little guidance and some simple steps, you'll be solving these like a pro in no time! So, grab your pencils and paper, and let's get started!

Understanding the Basics of Algebraic Equations

Before we jump into the solution, let's quickly recap the fundamental principles behind algebraic equations. At its core, an algebraic equation is a mathematical statement that asserts the equality of two expressions. These expressions can contain variables (like our 'y'), constants (like 3 and 9), and mathematical operations (like addition and subtraction). The main goal in solving an equation is to isolate the variable on one side of the equation to determine its value. This value is the solution to the equation, meaning it's the number that, when substituted for the variable, makes the equation true. Think of it like a puzzle – we need to figure out what number 'y' represents to make both sides of the equation balance.

In our equation, y + 3 = -y + 9, we have the variable 'y' appearing on both sides. This is a common scenario, and the key to solving it lies in understanding the concept of inverse operations. Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, and so are multiplication and division. We'll use these inverse operations to strategically move terms around in our equation until we get 'y' all by itself on one side.

Another crucial concept is maintaining the balance of the equation. Remember, the equals sign (=) represents a balance. Whatever operation we perform on one side of the equation, we must perform the exact same operation on the other side to keep the equation balanced. It's like a seesaw – if you add weight to one side, you need to add the same weight to the other side to keep it level. Failing to do so will lead to an incorrect solution.

Step-by-Step Solution: Solving y + 3 = -y + 9

Now, let's break down the solution to our equation y + 3 = -y + 9 step-by-step. We'll use inverse operations and maintain balance throughout the process.

Step 1: Combining 'y' Terms

Our first goal is to get all the 'y' terms on the same side of the equation. Looking at y + 3 = -y + 9, we see 'y' on the left side and '-y' on the right side. To eliminate the '-y' on the right, we'll use the inverse operation of subtraction, which is addition. We'll add 'y' to both sides of the equation:

y + 3 + y = -y + 9 + y

This simplifies to:

2y + 3 = 9

Notice how adding 'y' to both sides effectively canceled out the '-y' on the right side and combined with the 'y' on the left side to give us '2y'. We're one step closer to isolating 'y'!

Step 2: Isolating the 'y' Term

Now we have 2y + 3 = 9. To isolate the '2y' term, we need to get rid of the '+3' on the left side. Again, we'll use the inverse operation. The inverse of addition is subtraction, so we'll subtract 3 from both sides of the equation:

2y + 3 - 3 = 9 - 3

This simplifies to:

2y = 6

The '+3' and '-3' on the left side canceled each other out, leaving us with just '2y'. We're almost there!

Step 3: Solving for 'y'

Our final step is to get 'y' completely by itself. We currently have 2y = 6. This means '2' is being multiplied by 'y'. The inverse operation of multiplication is division, so we'll divide both sides of the equation by 2:

2y / 2 = 6 / 2

This simplifies to:

y = 3

And that's it! We've successfully solved the equation. The solution is y = 3. This means that if we substitute 3 for 'y' in the original equation, both sides will be equal.

Verifying the Solution

It's always a good practice to verify your solution to make sure you haven't made any mistakes along the way. To do this, we'll substitute our solution, y = 3, back into the original equation, y + 3 = -y + 9:

3 + 3 = -3 + 9

Simplifying both sides, we get:

6 = 6

Since both sides are equal, our solution y = 3 is correct! We've successfully navigated the algebraic seas and found our treasure.

Common Mistakes to Avoid

Solving equations can sometimes be tricky, and it's easy to make mistakes. Here are some common pitfalls to watch out for:

  • Forgetting to perform the same operation on both sides: This is the most crucial rule. If you add, subtract, multiply, or divide on one side, you must do the same on the other side to maintain balance.
  • Incorrectly applying inverse operations: Make sure you're using the correct inverse operation. Remember, addition and subtraction are inverses, and multiplication and division are inverses.
  • Combining unlike terms: You can only combine terms that have the same variable and exponent. For example, you can combine '2y' and 'y' to get '3y', but you can't combine '2y' and '3'.
  • Making arithmetic errors: Even a small arithmetic error can throw off your entire solution. Double-check your calculations to avoid these mistakes.

Practice Problems

Now that we've worked through one example, let's test your understanding with a few practice problems. Solving equations is like riding a bike – the more you practice, the better you'll get! Try solving these equations on your own:

  1. x - 5 = 2x + 1
  2. 3a + 7 = a - 3
  3. 2(z + 4) = 10

Remember to use the steps we discussed – combine like terms, isolate the variable term, and solve for the variable. And don't forget to verify your solutions!

Conclusion: Mastering the Art of Equation Solving

Solving equations is a fundamental skill in algebra and a valuable tool in many areas of mathematics and science. By understanding the basic principles, applying inverse operations, and maintaining balance, you can conquer even the most complex equations. Remember to practice regularly, learn from your mistakes, and don't be afraid to ask for help when you need it. With dedication and persistence, you'll become a master of equation solving!

So, there you have it! We've successfully tackled the equation y + 3 = -y + 9 and explored the key concepts behind solving algebraic equations. Keep practicing, and you'll be solving equations like a true mathematical whiz. Happy solving, guys!