Alright, let's break down this equation step-by-step to find the value of 'y'. This is a classic algebra problem, and we'll use basic arithmetic operations to isolate 'y' on one side of the equation. So, grab your pencil and paper, and let's dive in!

Step 1: Simplify the Right Side of the Equation

First, we need to simplify the right side of the equation: 2 * 37 + 2. To do this, we'll perform the multiplication first, according to the order of operations (PEMDAS/BODMAS).

So, what's 2 multiplied by 37? It's 74! Now we add 2 to that result:

2 * 37 = 74
74 + 2 = 76

Now our equation looks like this:

2y + 5 = 76

Step 2: Isolate the Term with 'y'

Our next goal is to isolate the term that contains 'y', which is 2y. To do this, we need to get rid of the + 5 on the left side of the equation. We can do this by subtracting 5 from both sides of the equation. Remember, whatever you do to one side of the equation, you have to do to the other side to keep it balanced!

2y + 5 - 5 = 76 - 5

This simplifies to:

2y = 71

Step 3: Solve for 'y'

Now we have 2y = 71. This means 2 times 'y' equals 71. To find the value of 'y', we need to divide both sides of the equation by 2:

2y / 2 = 71 / 2

This gives us:

y = 35.5

So, the value of 'y' is 35.5. You can express this as a decimal or as a fraction (71/2). Great job! You've successfully solved for 'y'. Remember, the key to solving these types of equations is to follow the order of operations and perform the same operations on both sides of the equation to maintain balance. Keep practicing, and you'll become a pro at algebra in no time!

Understanding Algebraic Equations

Algebraic equations, like the one we just solved, are fundamental tools in mathematics and are used extensively in various fields, from science and engineering to economics and computer science. Understanding how to solve them is a crucial skill. An algebraic equation is a statement that asserts the equality of two expressions. These expressions contain variables (like our 'y'), constants, and mathematical operations.

The goal of solving an algebraic equation is to find the value(s) of the variable(s) that make the equation true. This involves isolating the variable on one side of the equation by performing inverse operations. The order of operations (PEMDAS/BODMAS) is crucial in simplifying expressions and solving equations correctly.

Common Mistakes to Avoid

When solving algebraic equations, it's easy to make mistakes. Here are some common pitfalls to watch out for:

1. Not following the order of operations: Always remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Perform operations in the correct order to avoid incorrect simplifications.
2. Not applying operations to both sides: Whatever operation you perform on one side of the equation, you must perform the same operation on the other side to maintain the equality. For example, if you subtract 5 from the left side, you must also subtract 5 from the right side.
3. Incorrectly combining like terms: Make sure you only combine terms that are "like terms." Like terms have the same variable raised to the same power. For example, 3x and 5x are like terms, but 3x and 5x^2 are not.
4. Forgetting to distribute: When you have a term multiplied by an expression in parentheses, remember to distribute the term to each term inside the parentheses. For example, 2(x + 3) becomes 2x + 6.
5. Sign errors: Pay close attention to signs (positive and negative). A simple sign error can lead to an incorrect solution. Be especially careful when dealing with negative numbers and subtraction.

Tips for Success

Here are some tips to help you succeed in solving algebraic equations:

• Practice regularly: The more you practice, the better you'll become. Work through a variety of problems to build your skills and confidence.
• Show your work: Write down each step of your solution. This will help you keep track of your work and make it easier to identify any mistakes.
• Check your answer: Once you've found a solution, plug it back into the original equation to see if it makes the equation true. This is a great way to catch errors.
• Understand the concepts: Don't just memorize steps. Make sure you understand the underlying concepts. This will help you solve more complex problems and apply your knowledge in different situations.
• Ask for help: If you're struggling, don't be afraid to ask for help from a teacher, tutor, or classmate. Explaining the problem to someone else can also help you understand it better.

Real-World Applications

Solving algebraic equations isn't just a theoretical exercise. It has many real-world applications. Here are a few examples:

• Finance: Calculating interest rates, loan payments, and investment returns often involves solving algebraic equations.
• Physics: Many physics formulas are expressed as algebraic equations. For example, the equation d = vt (distance = velocity * time) can be solved for any of the variables if you know the other two.
• Engineering: Engineers use algebraic equations to design structures, analyze circuits, and solve a wide range of problems.
• Computer Science: Algebraic equations are used in algorithms, data analysis, and various other areas of computer science.

By mastering the skill of solving algebraic equations, you'll be well-equipped to tackle a wide range of problems in various fields. So keep practicing, stay curious, and enjoy the journey of learning!

Additional Practice Problems

To further hone your skills, here are a few more practice problems you can try:

1. Solve for x: 3x - 7 = 14
2. Solve for a: 5a + 9 = 2a - 3
3. Solve for z: 4(z + 2) = 20

Try solving these on your own, and then check your answers. Remember to show your work and follow the steps we discussed earlier. Good luck, and keep up the great work!

Conclusion

In summary, we successfully solved the equation 2y + 5 = 2 * 37 + 2 to find that y = 35.5. We walked through each step, simplifying the equation, isolating the variable, and performing the necessary operations to find the solution. We also discussed common mistakes to avoid, tips for success, and real-world applications of solving algebraic equations. By understanding these concepts and practicing regularly, you can become proficient in algebra and apply your knowledge to solve a wide range of problems.

Remember, algebra is a building block for more advanced math and science concepts, so mastering it now will set you up for success in the future. Keep practicing, stay curious, and don't be afraid to ask for help when you need it. You've got this! And remember, math can actually be fun when you approach it with the right attitude and a willingness to learn. Keep up the awesome work! You're doing great!