Solving Equations: A Step-by-Step Guide

Hey guys! Let's dive into a fun math problem today. We're gonna solve the equation 2(y - x) = x^2 - y + 6 given that x = 3. Don't worry if equations seem a little scary at first; we'll break it down step by step and make it super easy to understand. This is a common type of algebra problem, and mastering these skills is key to unlocking more advanced math concepts later on. So, grab your pencils and let's get started. Solving equations is like a puzzle; we're essentially trying to find the value of an unknown variable (in this case, y) that makes the equation true. We'll start by substituting the known value of x into the equation, then simplify and isolate y on one side of the equation.

Step 1: Substitute the Value of x

Our first step is to substitute the given value of x, which is 3, into the equation. Wherever you see x in the equation, you're going to replace it with 3. This gives us:

2(y - 3) = 3^2 - y + 6

See? It's that simple! We've taken our original equation and plugged in the value we know. This is a crucial step because it reduces the number of unknowns in our equation, bringing us closer to solving for y. The core idea here is to replace x with its numerical equivalent. This process is called substitution and is a fundamental technique in algebra. It allows us to transform equations into more manageable forms. Make sure you don't miss any instances of x when you're substituting. This often leads to errors. Think of substitution as replacing a placeholder (x) with its specific value (3).

Step 2: Simplify the Equation

Next, we need to simplify the equation. This involves performing the arithmetic operations. First, let's deal with the term 3^2. Remember that 3^2 means 3 multiplied by itself, which is 9. Also, we need to distribute the 2 on the left side of the equation. This means multiplying both y and -3 by 2.

So, our equation becomes:

2y - 6 = 9 - y + 6

Now, let's combine the constant terms on the right side of the equation. We have 9 and 6, which add up to 15. The equation then simplifies to:

2y - 6 = 15 - y

Simplifying is all about cleaning up the equation to make it easier to work with. Think of it like tidying up your room before you start studying; a cleaner space makes it easier to focus. We are following the order of operations (PEMDAS/BODMAS) to simplify the equation. This includes exponents, multiplication, addition, and subtraction. We are simplifying terms step by step to avoid confusion. Keep in mind to always perform operations within parentheses first, and then address the rest of the terms. Don't be afraid to take your time and double-check your calculations to prevent small mistakes.

Step 3: Isolate the Variable

Now, we need to get all the terms containing y on one side of the equation and the constant terms on the other side. Let's start by adding y to both sides of the equation. This will eliminate the - y on the right side. We have:

2y - 6 + y = 15 - y + y

Which simplifies to:

3y - 6 = 15

Next, let's eliminate the -6 from the left side by adding 6 to both sides:

3y - 6 + 6 = 15 + 6

This simplifies to:

3y = 21

Isolating the variable is where we bring everything together. The goal here is to get y by itself on one side of the equation. To do this, we perform inverse operations. Adding y to both sides eliminates the negative y and adding 6 to both sides eliminates the -6. It's like balancing a scale; anything we do to one side of the equation, we must do to the other to keep it equal. The order in which you perform these operations can sometimes change, but the ultimate goal is always to get y alone. Remember to keep an eye on the signs (positive or negative) of the terms. A simple mistake here can change your final answer.

Step 4: Solve for y

Finally, to solve for y, we need to get y completely alone. We have 3y = 21. To isolate y, we need to divide both sides of the equation by 3:

(3y) / 3 = 21 / 3

This gives us:

y = 7

And there you have it! We've found the solution to our equation. y equals 7. This means that when x is 3, the value of y that makes the original equation true is 7. We have successfully completed all the steps and arrived at our final answer. The key to this step is to isolate y. So, we divided both sides by the coefficient of y (which is 3 in this case). Division is the inverse operation of multiplication. Therefore, it helps us in isolating the unknown variable. Don't forget that whatever you do to one side of the equation, you must always do the same to the other side to keep the equation balanced.

Step 5: Verify Your Answer

It's always a good practice to verify your answer to make sure it's correct. We can do this by plugging the values of x and y back into the original equation and checking if both sides are equal.

Original equation: 2(y - x) = x^2 - y + 6

Substitute x = 3 and y = 7:

2(7 - 3) = 3^2 - 7 + 6

2(4) = 9 - 7 + 6

8 = 8

Since both sides are equal, our solution is correct! This is a super important step. Verify is similar to double-checking your work to prevent careless errors. By substituting back into the original equation, we're testing the accuracy of our calculated value. If both sides of the equation are equal, then your answer is most likely correct. If they are not equal, then you know there is a mistake somewhere along the way. Double-check all steps and calculations to find the error. Don't be afraid to revisit each step, paying special attention to signs, and calculations. You will then easily find the mistake. Always confirm your solution! This will help you enhance your accuracy.

Conclusion

So there you have it, guys! We successfully solved for y in the equation 2(y - x) = x^2 - y + 6, given that x = 3. By following these simple steps – substituting, simplifying, isolating the variable, and verifying – you can tackle similar algebra problems with confidence. Keep practicing, and you'll become a pro at solving equations in no time. Solving for equations builds a strong foundation for your journey in mathematics. Keep up the excellent work, and never stop learning! Remember, the more you practice, the easier these problems will become. Keep up the excellent work, and always remember to check your solutions!