Solving For Y In The Equation 7y - 6y - 10 = 13

Hey guys! Let's break down this math problem and solve for y step-by-step. We're tackling the equation $7y - 6y - 10 = 13$. Don't worry, it's much simpler than it looks! We will use order of operations (PEMDAS/BODMAS) and isolate the variable y on one side of the equation.

Understanding the Equation

Before we dive into solving for y, let’s make sure we understand what the equation is telling us. The equation $7y - 6y - 10 = 13$ involves a variable (y), some coefficients (the numbers in front of y), and constants (numbers without a variable). Our main goal here is to figure out what value of y will make the left side of the equation equal to the right side. Think of it like a puzzle where y is the missing piece!

The left side of the equation, $7y - 6y - 10$, has three terms. The first two terms, $7y$ and $-6y$, are like terms because they both contain the variable y. We can combine these terms to simplify the equation. The third term, $-10$, is a constant term. On the right side of the equation, we have the constant $13$. The equals sign (=) tells us that the expression on the left side has the same value as the expression on the right side.

To solve for y, we need to isolate it on one side of the equation. This means we want to manipulate the equation so that we have y all by itself on one side, and a number on the other side. We can do this by using inverse operations. Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations, and multiplication and division are inverse operations. We'll use these operations to get rid of the terms around y until we have y by itself.

Step-by-Step Solution

1. Combine Like Terms:

The first thing we want to do is simplify the left side of the equation by combining the y terms. We have $7y - 6y$. Think of this as having 7 y's and taking away 6 y's. How many y's are left? Just one! So, $7y - 6y = 1y$, which we can simply write as y. Our equation now looks like this:

$y - 10 = 13$

This step is crucial because it reduces the complexity of the equation, making it easier to handle. Combining like terms is a fundamental technique in algebra, and you'll use it often when solving equations. By simplifying the equation, we're one step closer to isolating y and finding its value.

2. Isolate y:

Now we need to get y by itself on the left side of the equation. We have $y - 10 = 13$. Notice that we have a $-10$ on the same side as y. To get rid of the $-10$, we need to perform the inverse operation. What's the inverse of subtraction? Addition! So, we'll add 10 to both sides of the equation. It’s super important to do the same thing to both sides to keep the equation balanced. Think of it like a scale – if you add something to one side, you need to add the same thing to the other side to keep it level.

$y - 10 + 10 = 13 + 10$

On the left side, the $-10$ and $+10$ cancel each other out, leaving us with just y. On the right side, $13 + 10$ equals 23. So, our equation now looks like this:

$y = 23$

And just like that, we've isolated y! This step demonstrates a key principle in solving equations: using inverse operations to undo operations and isolate the variable. By adding 10 to both sides, we effectively moved the constant term to the right side of the equation, leaving y alone on the left.

3. The Solution:

We've done it! We've solved for y. Our final answer is:

$y = 23$

This means that if we substitute 23 for y in the original equation, the equation will be true. To be absolutely sure, we can check our answer. Checking our solution is always a good idea, especially in more complex problems. It helps us catch any mistakes we might have made along the way.

Checking Our Answer

To check our answer, we'll substitute $y = 23$ back into the original equation:

$7y - 6y - 10 = 13$

Replace y with 23:

$7(23) - 6(23) - 10 = 13$

Now, let's do the math. First, we multiply:

$161 - 138 - 10 = 13$

Next, we subtract:

$23 - 10 = 13$

And finally:

$13 = 13$

Woo-hoo! The left side of the equation equals the right side, so our solution is correct. We can confidently say that $y = 23$ is the solution to the equation $7y - 6y - 10 = 13$. This check confirms that our step-by-step process was accurate, and we've successfully found the value of y.

Why This Matters

Solving for variables like y is a fundamental skill in algebra. It's not just about getting the right answer; it's about understanding how equations work and how to manipulate them. These skills are essential for more advanced math topics and have applications in many real-world situations, from calculating finances to designing buildings.

The process we used here – combining like terms and using inverse operations – is a powerful technique that you'll use again and again in algebra. The ability to break down a problem into smaller, manageable steps is also a crucial skill in mathematics and in life. So, by mastering these basic algebraic techniques, you're building a strong foundation for future success.

Real-World Applications

Algebra isn't just something you do in a math class; it's a tool for solving problems in the real world. For example, imagine you're planning a road trip. You need to figure out how much gas you'll need, how long the trip will take, and how much it will cost. These kinds of calculations often involve algebraic equations.

Or, let's say you're starting a small business. You need to calculate your costs, your revenue, and your profit. Algebra can help you create a budget, forecast your earnings, and make informed decisions about your business. From cooking recipes to managing investments, algebra is a valuable skill in many aspects of life.

Tips for Success

Here are a few tips to help you succeed in solving equations:

• Practice, practice, practice: The more you practice, the better you'll become at solving equations. Do lots of problems, and don't be afraid to make mistakes. Mistakes are learning opportunities!
• Show your work: Write down each step as you solve the equation. This will help you keep track of your work and make it easier to find mistakes.
• Check your answers: Always check your answers by substituting them back into the original equation. This will help you catch errors and build confidence in your solutions.
• Ask for help: If you're struggling with a problem, don't be afraid to ask for help. Talk to your teacher, a tutor, or a friend. There are also many online resources that can help you learn algebra.

Conclusion

So, there you have it! We've successfully solved for y in the equation $7y - 6y - 10 = 13$. Remember, the key is to combine like terms and use inverse operations to isolate the variable. Keep practicing, and you'll become an algebra pro in no time! You got this, guys!