Solving 4y - 7 = 5 A Step-by-Step Guide

Have you ever been faced with a mathematical equation and felt a slight sense of intimidation? Well, fret no more! In this article, we're going to break down the process of solving a simple algebraic equation step-by-step. Our focus will be on the equation 4y - 7 = 5, a classic example that's perfect for beginners and a great refresher for those with some experience. We'll take a conversational approach, ensuring you understand not just the 'how' but also the 'why' behind each step. So, whether you're a student tackling homework, a professional brushing up on your skills, or simply someone curious about math, this guide is for you. Let's dive in and unravel this equation together!

Understanding the Basics of Algebraic Equations

Before we jump into the nitty-gritty of solving algebraic equations, let's take a moment to understand what they really are. Think of an equation as a balanced scale. On one side, you have an expression, and on the other, you have another expression. The equals sign (=) is the fulcrum, ensuring both sides hold the same value. In our case, the equation 4y - 7 = 5 tells us that the expression "4y - 7" has the same value as the number 5. The goal of solving an equation is to isolate the variable, which in our case is 'y'. This means we want to get 'y' all by itself on one side of the equation so we can see what value it must have to make the equation true. Variables are like mystery boxes; they represent an unknown number we're trying to discover. To isolate 'y', we'll use a series of algebraic operations. These operations are like adding or removing weights from our balanced scale, but with a crucial rule: whatever we do to one side, we must do to the other to maintain the balance. This principle of equality is the cornerstone of solving equations. We will delve into this more as we progress, showcasing how addition, subtraction, multiplication, and division play their roles in solving for 'y'. So, keep this balanced scale analogy in mind as we proceed; it’s a simple yet powerful way to visualize what we're doing in algebra.

Step-by-Step Solution to 4y - 7 = 5

Alright, guys, let’s get our hands dirty and solve the equation 4y - 7 = 5! We'll break it down into simple, manageable steps. Remember our balanced scale analogy? We're aiming to get 'y' alone on one side while keeping the equation balanced.

Step 1: Isolating the Term with 'y'

The first thing we want to do is isolate the term that contains our variable, 'y'. In this case, that term is "4y". Notice that we have a "- 7" on the same side as "4y". To get rid of this "- 7", we'll perform the opposite operation, which is adding 7. But remember, whatever we do to one side, we must do to the other to keep the equation balanced. So, we add 7 to both sides:

4y - 7 + 7 = 5 + 7

On the left side, "- 7 + 7" cancels out, leaving us with just "4y". On the right side, "5 + 7" equals 12. Our equation now looks like this:

4y = 12

We've successfully isolated the term with 'y'. We're one step closer to finding the value of 'y'. It's like we're peeling away the layers to reveal the mystery variable underneath!

Step 2: Solving for 'y'

Now that we have 4y = 12, the next step is to isolate 'y' completely. Currently, 'y' is being multiplied by 4. To undo this multiplication, we'll perform the inverse operation, which is division. Just like before, we need to do the same thing to both sides of the equation to maintain balance. So, we divide both sides by 4:

4y / 4 = 12 / 4

On the left side, "4y / 4" simplifies to 'y', as the 4s cancel each other out. On the right side, "12 / 4" equals 3. Our equation now looks like this:

y = 3

And there we have it! We've solved for 'y'. The value of 'y' that makes the equation 4y - 7 = 5 true is 3. It's like we've cracked the code and revealed the hidden number. But, to be absolutely sure, we should always check our answer.

Step 3: Checking the Solution

Checking our solution is a crucial step in algebra. It's like proofreading your work before submitting it. To check if y = 3 is the correct solution, we'll substitute 3 back into the original equation:

4y - 7 = 5

Replace 'y' with 3:

4(3) - 7 = 5

Now, we simplify the left side of the equation. First, we multiply 4 by 3, which gives us 12:

12 - 7 = 5

Next, we subtract 7 from 12, which gives us 5:

5 = 5

The left side of the equation now equals the right side. This confirms that our solution, y = 3, is indeed correct! It's like the final piece of the puzzle clicking into place. We've not only solved the equation, but we've also verified our answer, giving us complete confidence in our solution.

Common Mistakes to Avoid

When solving algebraic equations, there are a few common pitfalls that students often encounter. Recognizing these mistakes can save you from unnecessary errors and help you build a stronger understanding of algebra. Let’s take a look at some of these common blunders and how to steer clear of them.

Mistake 1: Forgetting to Apply Operations to Both Sides

One of the most frequent errors is failing to perform the same operation on both sides of the equation. Remember our balanced scale analogy? If you add or subtract something from one side without doing the same to the other, you'll throw off the balance and the equation will no longer be true. For example, in the equation 4y - 7 = 5, if you add 7 only to the left side, you'll end up with 4y = 5, which is incorrect. Always remember, whatever you do to one side, you must do to the other.

Mistake 2: Incorrect Order of Operations

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), is crucial in simplifying expressions. Failing to follow this order can lead to incorrect answers. For instance, if you were to subtract 7 from 4y before adding 7 to both sides in our equation, you'd be going against the correct order of operations and would likely end up with the wrong solution. Always simplify expressions according to PEMDAS.

Mistake 3: Sign Errors

Sign errors, especially with negative numbers, are another common source of mistakes. When adding, subtracting, multiplying, or dividing negative numbers, it's easy to make a mistake if you're not careful. For example, incorrectly handling the negative sign when moving terms across the equals sign can lead to an incorrect solution. Pay close attention to the signs and double-check your work to avoid these errors.

Mistake 4: Not Checking the Solution

As we discussed earlier, checking your solution is an essential step. Skipping this step means you might not catch a mistake you made during the solving process. By substituting your solution back into the original equation, you can verify its correctness and gain confidence in your answer. Make it a habit to always check your solution.

Mistake 5: Combining Unlike Terms

Combining unlike terms is a common algebraic error. You can only combine terms that have the same variable and exponent. For example, you cannot combine 4y and 7 because they are not like terms. Similarly, you can't combine terms with different exponents, like y and y². Always ensure you're combining only like terms to avoid this mistake.

By being aware of these common mistakes and taking the time to avoid them, you'll improve your accuracy and confidence in solving algebraic equations. Remember, practice makes perfect, so keep working at it and you'll become a pro in no time!

Practice Problems

To truly master the art of solving equations, practice is key. Working through various problems will help solidify your understanding and improve your problem-solving skills. Let's put our newfound knowledge to the test with a few practice problems similar to the one we've just solved. Remember to follow the steps we discussed: isolate the term with the variable, solve for the variable, and always check your solution. Grab a pencil and paper, and let's get started!

Practice Problem 1

Solve for 'x':

2x + 5 = 11

Practice Problem 2

Solve for 'a':

3a - 8 = 7

Practice Problem 3

Solve for 'm':

5m + 2 = 17

Practice Problem 4

Solve for 'z':

6z - 4 = 20

Practice Problem 5

Solve for 'k':

4k + 9 = 1

Take your time to work through each problem, applying the steps we've learned. Remember, the goal is not just to find the answer, but to understand the process. Once you've solved these problems, you can check your answers against the solutions provided below. If you get stuck, don't hesitate to revisit the steps we discussed earlier in the article. And remember, every mistake is an opportunity to learn and improve. Happy solving!

Solutions to Practice Problems

Ready to check your answers and see how you did on the practice problems? Here are the solutions, worked out step-by-step, just like we did with our original equation. Compare your solutions to these, and if you made any mistakes, take a moment to understand where you went wrong. This is a crucial part of the learning process. So, let's dive in and review the solutions together!

Solution to Practice Problem 1: 2x + 5 = 11

1. Subtract 5 from both sides:

   2x + 5 - 5 = 11 - 5
   2x = 6
   

2. Divide both sides by 2:

   2x / 2 = 6 / 2
   x = 3
   

   Solution: x = 3

Solution to Practice Problem 2: 3a - 8 = 7

1. Add 8 to both sides:

   3a - 8 + 8 = 7 + 8
   3a = 15
   

2. Divide both sides by 3:

   3a / 3 = 15 / 3
   a = 5
   

   Solution: a = 5

Solution to Practice Problem 3: 5m + 2 = 17

1. Subtract 2 from both sides:

   5m + 2 - 2 = 17 - 2
   5m = 15
   

2. Divide both sides by 5:

   5m / 5 = 15 / 5
   m = 3
   

   Solution: m = 3

Solution to Practice Problem 4: 6z - 4 = 20

1. Add 4 to both sides:

   6z - 4 + 4 = 20 + 4
   6z = 24
   

2. Divide both sides by 6:

   6z / 6 = 24 / 6
   z = 4
   

   Solution: z = 4

Solution to Practice Problem 5: 4k + 9 = 1

1. Subtract 9 from both sides:

   4k + 9 - 9 = 1 - 9
   4k = -8
   

2. Divide both sides by 4:

   4k / 4 = -8 / 4
   k = -2
   

   Solution: k = -2

How did you do? Don't worry if you didn't get all the answers right. The most important thing is that you're practicing and learning. If you consistently missed a particular step, go back and review that section of the article. Remember, solving equations is a skill that improves with practice. Keep at it, and you'll become more confident and proficient in no time!

Conclusion

Congratulations, guys! You've successfully navigated the process of solving the equation 4y - 7 = 5. We've covered the fundamental principles of algebraic equations, walked through a step-by-step solution, discussed common mistakes to avoid, and even put our skills to the test with practice problems. Solving equations like this is a foundational skill in mathematics, and it opens the door to more complex concepts and applications. Remember, the key to mastering algebra, like any skill, is consistent practice and a willingness to learn from your mistakes. So, keep practicing, stay curious, and don't be afraid to tackle new challenges. Whether you're solving equations for school, work, or personal enrichment, the ability to think logically and systematically will serve you well in all aspects of life. We hope this guide has been helpful and has boosted your confidence in solving algebraic equations. Keep up the great work, and happy solving!