Solving For Y: A Step-by-Step Guide To 2(4y + 5) = 34

Oct 25, 2025 by Andrew McMorgan 54 views

Hey math enthusiasts! Ever find yourself staring at an equation and wondering, "Where do I even start?" Don't sweat it! We're going to break down one of those equations today, making it super clear and easy to understand. We're tackling how to solve for y in the equation 2(4y + 5) = 34. So grab your pencils, and let's dive in!

Understanding the Basics of Algebraic Equations

Before we jump straight into the problem, let's quickly touch on some fundamental concepts in algebra. Understanding these basics is like having the right tools in your toolbox – they make the whole job easier!

What is an Equation? At its core, an equation is a mathematical statement that shows two expressions are equal. Think of it as a balanced scale; what's on one side must be equal to what's on the other. The equals sign (=) is the key here, indicating this balance. So, $2(4y + 5) = 34$ is telling us that whatever the value of the expression on the left side (2 multiplied by 4y + 5) is, it must be the same as the value on the right side (which is 34).
The Goal: Isolating the Variable In most algebraic equations, we're trying to find the value of an unknown, often represented by a letter like 'y', 'x', or 'z'. This unknown is called a variable. Solving an equation essentially means finding out what number this variable represents. To do that, our main goal is to isolate the variable. Isolation means getting the variable alone on one side of the equation. When we have 'y = some number', we've successfully solved for y. Think of it like a treasure hunt – the variable is the treasure, and isolating it is finding it! We do this by using inverse operations, which we'll talk about next.
Inverse Operations: The Key to Isolating Inverse operations are operations that "undo" each other. They are essential tools for isolating variables. Here are some key pairs of inverse operations:
- Addition and Subtraction: If an equation has something added to the variable (like y + 3), we use subtraction to undo it (subtracting 3 from both sides). Conversely, if something is subtracted (like y - 5), we use addition (adding 5 to both sides).
- Multiplication and Division: If the variable is multiplied by a number (like 4y), we use division to undo it (dividing both sides by 4). If the variable is divided by a number (like y/2), we use multiplication (multiplying both sides by 2).
The Golden Rule of Equations: Keep it Balanced! This is the most important rule when solving equations. Whatever operation you perform on one side of the equation, you must perform the exact same operation on the other side. This is because we need to maintain the balance. If you add 5 to one side but not the other, the equation is no longer equal. Think back to our balanced scale analogy – if you add weight to one side, you must add the same weight to the other to keep it balanced. This principle is what allows us to manipulate equations and isolate the variable while maintaining the equality.

With these basics under our belt, we're ready to tackle our specific equation! Remember, algebra can seem daunting at first, but with practice, it becomes much more intuitive. We'll apply these concepts step-by-step, so you can see exactly how it works in action.

Step-by-Step Solution for 2(4y + 5) = 34

Alright, guys, let's get into the nitty-gritty of solving this equation. We're going to break it down step by step, so you can follow along easily. Remember our goal: to isolate 'y'.

Step 1: Distribute the 2

When you see parentheses in an equation, especially with a number right outside them, the first thing you usually want to do is distribute. Distributing means multiplying the number outside the parentheses by each term inside the parentheses. In our equation, 2(4y + 5) = 34, we need to distribute the 2 to both the 4y and the 5.

So, let's do it:

2 * 4y = 8y
2 * 5 = 10

Now, rewrite the equation with the distribution done:

8y + 10 = 34

See? We've already made progress! The parentheses are gone, and the equation looks a bit simpler. Distributing is a crucial step in many algebraic equations, so getting comfortable with it is a big win.

Step 2: Isolate the Term with 'y'

Now that we've distributed, our goal is to isolate the term with 'y' in it – which is 8y in this case. Remember, isolating means getting it alone on one side of the equation. Right now, we have "8y + 10". That + 10 is what's keeping the 8y from being completely alone.

To get rid of the + 10, we need to use the inverse operation. The inverse of addition is subtraction. So, we're going to subtract 10 from both sides of the equation. Remember the golden rule: what we do to one side, we must do to the other to keep the equation balanced.

Here's how it looks:

8y + 10 - 10 = 34 - 10

On the left side, the +10 and -10 cancel each other out, leaving us with just 8y. On the right side, 34 - 10 equals 24. So, our equation now looks like this:

8y = 24

We're getting closer! The term with 'y' is now isolated on the left side. Just one more step to go!

Step 3: Solve for 'y'

We're in the home stretch! We have 8y = 24, and we want to find out what just one 'y' equals. Right now, 'y' is being multiplied by 8. To isolate 'y' completely, we need to undo this multiplication.

The inverse operation of multiplication is division. So, we're going to divide both sides of the equation by 8. Again, we're keeping that equation balanced!

Here's the math:

8y / 8 = 24 / 8

On the left side, the 8s cancel each other out, leaving us with just 'y'. On the right side, 24 divided by 8 is 3. So, we have:

y = 3

We did it! We've solved for 'y'! The value of 'y' that makes the equation 2(4y + 5) = 34 true is 3.

Step 4: Check Your Answer

Okay, we think we've got the answer, but it's always a good idea to double-check, especially in math. Checking your answer is like having a safety net – it helps you catch any mistakes.

To check our answer, we're going to substitute the value we found for 'y' (which is 3) back into the original equation: 2(4y + 5) = 34. If our answer is correct, both sides of the equation should be equal when we plug in y = 3.

Let's do it:

2(4 * 3 + 5) = 34

Now, we simplify following the order of operations (PEMDAS/BODMAS):

First, deal with the parentheses: 4 * 3 = 12, so we have 2(12 + 5) = 34
Continue inside the parentheses: 12 + 5 = 17, so we have 2(17) = 34
Now, multiply: 2 * 17 = 34

So, we have 34 = 34. The left side equals the right side! This means our solution, y = 3, is correct. Woo-hoo!

Common Mistakes to Avoid

Solving equations can be tricky, and it's easy to slip up if you're not careful. Here are some common mistakes people make when solving equations, so you can be extra aware and avoid them:

Forgetting to Distribute Properly: When you have a number outside parentheses, you need to multiply it by every term inside. A common mistake is to multiply by the first term but forget the second (or third, etc.). Always double-check that you've distributed correctly.
Not Applying Operations to Both Sides: Remember the golden rule! Whatever you do to one side of the equation, you must do to the other. If you subtract 5 from the left side, you have to subtract 5 from the right side too. Failing to do this throws the equation out of balance and leads to the wrong answer.
Incorrectly Combining Like Terms: Like terms are terms that have the same variable raised to the same power (or are just constants). You can only combine like terms. For example, you can combine 3x and 5x to get 8x, but you can't combine 3x and 5. Mixing up like and unlike terms is a common source of errors.
Order of Operations Errors: Remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). You need to perform operations in the correct order. If you add before you multiply, for instance, you'll get the wrong answer.
Sign Errors: Watch out for those negative signs! They can be sneaky. Make sure you're applying the rules of signed numbers correctly (e.g., a negative times a negative is a positive). A simple sign error can throw off the whole solution.
Skipping Steps: It might be tempting to skip steps to save time, but this can often lead to mistakes. Writing out each step clearly helps you keep track of what you're doing and reduces the chance of errors. Especially when you're learning, it's best to be thorough.

By being aware of these common pitfalls, you can minimize your chances of making mistakes and solve equations with more confidence.

Practice Problems

Okay, now that we've walked through an example and talked about common mistakes, it's time to put your skills to the test! Practice is key to mastering algebra. Here are a few practice problems for you to try. Grab a pencil and paper, and let's get solving!

Solve for x: 3(2x - 1) = 21
Solve for a: 5a + 8 = 23
Solve for m: -2(3m + 4) = -10
Solve for z: 4z - 7 = 5
Solve for p: 6(p - 2) = 18

(Answers will be provided at the end of this section, so try to solve it yourself first before looking at the answer)

For 1: x = 4
For 2: a = 3
For 3: m = 1/3
For 4: z = 3
For 5: p = 5

These problems are similar to the one we just solved, so you can use the same steps as a guide. Remember to distribute first if there are parentheses, then isolate the variable term, and finally, solve for the variable. Don't forget to check your answers!

If you get stuck, don't worry! That's perfectly normal. Go back and review the steps we discussed earlier. Identify the step where you're having trouble, and focus on that. Maybe you need to brush up on distribution, or perhaps you're having trouble with inverse operations. Whatever it is, pinpointing the problem area is the first step to overcoming it.

Conclusion

Alright, guys, we've covered a lot in this guide! We've walked through how to solve the equation 2(4y + 5) = 34 step-by-step. We started with the basics of algebraic equations, emphasizing the importance of isolating the variable and the golden rule of keeping the equation balanced. We then tackled our specific equation, breaking down each step: distributing, isolating the 'y' term, solving for 'y', and checking our answer. Remember, checking your answer is a crucial step in the problem-solving process to ensure you have the correct answer.

We also discussed common mistakes to avoid, such as forgetting to distribute properly, not applying operations to both sides, and making sign errors. Being aware of these potential pitfalls will help you solve equations more accurately and confidently. We also provided practice problems for you to hone your skills. Remember, practice makes perfect! The more you solve equations, the more comfortable and proficient you'll become.

Solving algebraic equations is a fundamental skill in mathematics. It's not just about getting the right answer; it's about developing your problem-solving skills and logical thinking. These skills are valuable not only in math class but also in many other areas of life.

So, keep practicing, stay patient, and don't be afraid to ask for help when you need it. You've got this! Now go out there and conquer those equations!