Let's break down this equation and find out what 'y' is! It looks a bit intimidating at first, but don't worry, we'll solve it step by step. So, grab a pen and paper (or your favorite note-taking app) and let's dive in!

Understanding the Equation

First, let's rewrite the equation to make it a bit clearer:

2y + 5 = 2 * 37 + 2

What this means is that we have a variable 'y' that we need to isolate. We want to get 'y' all by itself on one side of the equation so we know its value. The left side of the equation involves 'y' multiplied by 2, plus 5. The right side involves multiplying 2 by 37, and then adding 2. Our job is to manipulate the equation until we know what 'y' equals.

The Order of Operations

Remember PEMDAS (or BODMAS)? It's the order of operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This is super important because we need to simplify the right side of the equation first before we start moving things around to isolate 'y'.

Simplifying the Right Side

Okay, let's focus on the right side: 2 * 37 + 2. According to PEMDAS, we need to do the multiplication before the addition. So:

2 * 37 = 74

Now, add 2:

74 + 2 = 76

So, the right side of the equation simplifies to 76. Now our equation looks like this:

2y + 5 = 76

Isolating 'y'

Now comes the fun part – getting 'y' all by itself. To do this, we need to undo the operations that are being done to 'y'. Right now, 'y' is being multiplied by 2, and then 5 is being added. We'll undo these in reverse order.

Subtracting 5 from Both Sides

The opposite of adding 5 is subtracting 5. We need to subtract 5 from both sides of the equation to keep it balanced. Imagine the equation as a scale – if you do something to one side, you have to do the same thing to the other side to keep it even.

So, subtract 5 from both sides:

2y + 5 - 5 = 76 - 5

This simplifies to:

2y = 71

Dividing by 2

Now we have 2y = 71. 'y' is being multiplied by 2. To undo this, we need to divide both sides by 2:

2y / 2 = 71 / 2

This simplifies to:

y = 71 / 2

The Solution

So, y = 71 / 2. If you want to express this as a decimal, you can divide 71 by 2:

y = 35.5

Therefore, the value of y is 35.5.

Let's Recap

To solve the equation 2y + 5 = 2 * 37 + 2, we followed these steps:

1. Simplified the right side of the equation using the order of operations.
2. Subtracted 5 from both sides to isolate the term with 'y'.
3. Divided both sides by 2 to solve for 'y'.

And that's it! We found that y = 35.5. Great job! Equations like these might seem tricky at first, but with a little practice, you'll be solving them like a pro in no time. Keep practicing, and don't be afraid to ask for help if you get stuck. Happy solving! Remember that math is like building with blocks, each concept builds on the previous, and solving for variables is a key skill to understand more advanced mathematics. By understanding each step, it makes it easier to solve other equation types.

Why is this important?

Solving for variables is one of the most fundamental skills in algebra and mathematics in general. Mastering this technique allows us to solve a wide range of problems in various fields such as science, engineering, economics, and computer science. It's not just about finding a number; it's about understanding relationships and representing them mathematically.

Real-World Applications

Think about situations in daily life where you might need to solve for a variable. For example, if you are planning a road trip and want to calculate how much gas you'll need, you might use an equation to relate distance, gas mileage, and the amount of gas required. Similarly, if you're cooking and need to adjust a recipe for a different number of servings, you might use an equation to scale the ingredients. In finance, understanding how interest rates affect investments requires solving for variables in financial formulas.

Building Blocks for Advanced Math

Understanding how to solve for variables is a crucial stepping stone to more advanced mathematical concepts such as calculus, differential equations, and linear algebra. These fields build upon the basic principles of algebra and are essential tools for modeling and solving complex problems in science and engineering.

Critical Thinking and Problem-Solving

The process of solving for variables cultivates critical thinking and problem-solving skills. It requires you to analyze a situation, identify the relevant variables, and apply the appropriate mathematical operations to find the unknown value. These skills are valuable not only in mathematics but also in other areas of life where you need to make decisions based on available information.

Tips for Solving Equations Like a Pro

1. Simplify First

Before you start isolating the variable, simplify both sides of the equation as much as possible. Combine like terms, perform any necessary multiplications or divisions, and get rid of any unnecessary clutter. This will make the equation easier to work with and reduce the chances of making mistakes.

2. Isolate the Variable

The goal is to get the variable all by itself on one side of the equation. To do this, use inverse operations to undo whatever is being done to the variable. Remember to perform the same operation on both sides of the equation to keep it balanced.

3. Check Your Work

Once you have found a solution, plug it back into the original equation to make sure it works. If the equation holds true, then you have found the correct solution. If not, go back and check your steps to see where you made a mistake.

4. Practice, Practice, Practice

The more you practice solving equations, the better you will become at it. Start with simple equations and gradually work your way up to more complex ones. Don't be afraid to make mistakes – they are part of the learning process. Keep at it, and you'll be solving equations like a pro in no time!

5. Understand the 'Why'

It's not just about memorizing steps; it's about understanding the underlying principles. When you understand why you are doing something, it becomes much easier to remember and apply the techniques in different situations. Always ask yourself, "Why am I doing this step?" and "How does this help me solve the problem?"

By understanding the importance of solving for variables and following these tips, you'll be well on your way to mastering this essential mathematical skill. Keep exploring, keep practicing, and never stop learning! You got this!