Solving Equations: A Step-by-Step Guide

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Hey math enthusiasts! Let's dive into solving equations, specifically the one you've tossed our way: Solve for y. -3y + 22 = 2(y - 4). Don't worry, it might look a bit intimidating at first, but trust me, it's totally manageable. We'll break it down into easy-to-follow steps, making sure you grasp every concept along the way. Get ready to flex those math muscles and feel the satisfaction of cracking the code! So, the goal is to find the value of 'y' that makes this equation true. Think of it like a puzzle – we need to rearrange the pieces until we isolate 'y' and reveal its secret identity. Are you ready?

First things first, understanding the equation. We've got variables, constants, and operations all mixed together. On the left side, we have -3y + 22. This means we're multiplying 'y' by -3 and then adding 22. On the right side, we have 2(y - 4). The parentheses tell us that we need to first subtract 4 from 'y' and then multiply the result by 2. It’s like a recipe where we have to follow specific steps in a particular order to get the right outcome. The equals sign (=) is the balance point, showing us that both sides of the equation have the same value. Our job is to keep that balance while we manipulate the equation to isolate 'y'.

Now, let's roll up our sleeves and get to work. Our first step will be to distribute the 2 on the right side of the equation. Distributing means multiplying the number outside the parentheses (2 in this case) by each term inside the parentheses. So, 2 times 'y' is 2y, and 2 times -4 is -8. This simplifies the right side to 2y - 8. Our equation now looks like this: -3y + 22 = 2y - 8. We’ve removed the parentheses, and the equation is a little less cluttered and easier to deal with. This is all about making the equation simpler so we can solve for 'y'.

Next, let’s gather all the 'y' terms on one side of the equation. To do this, we can add 3y to both sides. Remember, whatever we do to one side of the equation, we must do to the other side to keep it balanced. Adding 3y to the left side cancels out the -3y, leaving us with just 22. Adding 3y to the right side combines with 2y to give us 5y. Our equation is now: 22 = 5y - 8. We’re getting closer to isolating 'y'!

Now, to continue simplifying, we want to isolate the term with 'y'. Our next step is to add 8 to both sides. Adding 8 to the left side, 22 + 8, gives us 30. Adding 8 to the right side cancels out the -8, leaving us with just 5y. So now the equation looks like this: 30 = 5y. The equation is getting easier, right? We're isolating the variable 'y' to find its value.

Isolating the Variable

Alright, folks, we're in the home stretch now! We've made great progress by simplifying the equation. Now, we are trying to find the value of 'y'. To isolate 'y', we need to get rid of the 5 that is currently multiplying it. How do we do that? By dividing both sides by 5. This is the final step to find the value of the variable. Dividing 30 by 5 gives us 6, and dividing 5y by 5 leaves us with just 'y'. Therefore, 6 = y, or as we usually write it, y = 6. Tada! We've successfully solved for 'y'! The solution is y = 6. This is the final value of the variable. By following these steps, we've found the solution. Amazing, isn't it? Equations can be fun, too, when you master them!

So, to recap, the steps were:

  1. Distribute the 2 on the right side. -3y + 22 = 2y - 8
  2. Add 3y to both sides. 22 = 5y - 8
  3. Add 8 to both sides. 30 = 5y
  4. Divide both sides by 5. y = 6

Verifying Your Solution

Once we think we've found the solution, it's always a good idea to check our answer. This is like double-checking your work to make sure you didn't miss anything. We do this by plugging the value of 'y' (which is 6) back into the original equation and seeing if both sides are equal. Let's do it:

Original equation: -3y + 22 = 2(y - 4)

Substitute y = 6: -3(6) + 22 = 2(6 - 4)

Simplify: -18 + 22 = 2(2)

Continue simplifying: 4 = 4

Because both sides of the equation are equal (4 = 4), we know our solution, y = 6, is correct! High five, we did it!

This verification step is crucial. It confirms that the value we found for 'y' actually makes the original equation true. If the sides weren't equal, we'd know we made a mistake somewhere and need to go back and check our work. This practice not only ensures accuracy but also reinforces our understanding of equations. It builds our confidence and makes us better problem-solvers. We're not just solving; we are learning the ability to solve the equation. This is the main goal.

More Practice with Equations

Want to become an equation-solving pro? The best way is to practice, practice, practice! Here are a few more equations for you to try. Remember to follow the steps we covered and don't be afraid to make mistakes – that's how we learn. Each time we try a new equation, we're reinforcing our skills and building our confidence. So let’s have a look at them:

  1. 4x + 7 = 19
  2. 2(x + 3) = 10
  3. 5x - 3 = 2x + 9

Try these on your own. You can check your answers by substituting the values back into the equations. If you find yourself struggling, go back and review the steps we've outlined. Remember, solving equations is a skill that improves with practice. The more you work with equations, the more familiar you’ll become with the process. We encourage you to seek more resources or ask for help. Don’t be afraid to ask questions. Every question you ask and every equation you solve will boost your confidence and make you a more confident math problem-solver.

Now, let's briefly touch on some important tips and tricks that can help you solve equations more efficiently:

  • Always follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). Doing so, you won't have any issues.
  • Keep your work organized: Write each step clearly and neatly. This will help you avoid errors and make it easier to find mistakes if you get stuck. Think of it as a clear roadmap.
  • Double-check your answers: Substituting your solution back into the original equation is an excellent habit. It's an easy way to verify your work and catch any errors. That way, you won't have any problem with the equation.

As you practice these steps and master the tricks, solving equations will feel less intimidating. You'll gain a deeper understanding of the relationships between variables, constants, and operations. This newfound knowledge will be valuable not only in your math classes but also in various aspects of life. You may not realize it, but solving equations is a type of critical thinking and problem-solving, which is essential to many aspects of life. And remember, the key to success is to stay persistent and never give up. Keep solving, keep learning, and keep growing! You've got this!