Solving For Y: A Step-by-Step Guide To 7 = 7(5 + 3y) - 19y
Hey guys! Let's dive into solving a classic algebraic equation. Today, we're tackling the equation 7 = 7(5 + 3y) - 19y. Don't worry, it might look a little intimidating at first, but we'll break it down step by step, making it super easy to understand. So, grab your pencils and let's get started!
Understanding the Equation
Before we jump into solving, let's quickly understand what the equation is asking us. We have an equation where 'y' is an unknown variable. Our goal is to isolate 'y' on one side of the equation to find its value. We'll do this by using algebraic operations – things like distributing, combining like terms, and using inverse operations.
Remember, equations are like a balanced scale. What we do to one side, we must do to the other to keep it balanced. This principle is key to solving for 'y' accurately. So, always keep this balance in mind as we go through each step. The most important thing is to make sure that each step you take is accurate.
When dealing with such mathematical problems, it’s essential to pay attention to detail. A small mistake in one step can lead to a wrong answer. Therefore, it's always a good idea to double-check your work as you go along. Also, understanding the order of operations (PEMDAS/BODMAS) is crucial to correctly simplify expressions. In our case, we'll start by dealing with the parentheses and then move on to other operations. So, let's get started and see how we can crack this equation!
Step 1: Distribute the 7
The first thing we need to do is simplify the right side of the equation. We have 7(5 + 3y). To get rid of these parentheses, we'll use the distributive property. This means we multiply the 7 by both the 5 and the 3y inside the parentheses.
So, 7 multiplied by 5 is 35. And 7 multiplied by 3y is 21y. Now our equation looks like this:
7 = 35 + 21y - 19y
See how much cleaner that looks already? We've taken the first step in untangling this equation. Distributing is a fundamental skill in algebra, and it's something you'll use all the time, so it's great to get comfortable with it. This step essentially expands the expression, making it easier to combine like terms later on. Make sure to double-check your multiplication to avoid any errors. Alright, let's move on to the next step and continue our journey to solving for 'y'!
Step 2: Combine Like Terms
Now that we've distributed, let's simplify the right side of the equation further. We have 35 + 21y - 19y. Notice that we have two terms with 'y' in them: 21y and -19y. These are like terms, which means we can combine them.
To combine them, we simply add their coefficients (the numbers in front of the 'y'). So, 21y minus 19y is 2y. Our equation now looks like this:
7 = 35 + 2y
Isn't it getting simpler? Combining like terms is a powerful technique in algebra. It helps to clean up the equation and make it easier to isolate the variable we're trying to solve for. This step reduces the number of terms in the equation, making subsequent steps less complicated. Remember, like terms are terms that have the same variable raised to the same power. Once you've identified the like terms, combining them is usually a straightforward addition or subtraction. Let's keep the momentum going and move on to the next step!
Step 3: Isolate the 'y' Term
Our next goal is to get the term with 'y' (which is 2y) all by itself on one side of the equation. Currently, we have 7 = 35 + 2y. We need to get rid of that 35 on the right side.
To do this, we'll use the inverse operation. Since 35 is being added, we'll subtract 35 from both sides of the equation. Remember, what we do to one side, we must do to the other to keep the equation balanced.
So, subtracting 35 from both sides gives us:
7 - 35 = 35 + 2y - 35
This simplifies to:
-28 = 2y
We're getting closer! We've successfully isolated the term with 'y'. This step is crucial because it sets us up to finally solve for 'y' in the next step. Isolating the variable term often involves using addition or subtraction to move constants to the other side of the equation. It's like peeling away layers to reveal the value of 'y'. Keep up the great work, guys; we're almost there!
Step 4: Solve for 'y'
We're in the home stretch! We now have -28 = 2y. To finally solve for 'y', we need to get 'y' all by itself. Currently, 'y' is being multiplied by 2.
To undo this multiplication, we'll use the inverse operation: division. We'll divide both sides of the equation by 2.
So, dividing both sides by 2 gives us:
-28 / 2 = 2y / 2
This simplifies to:
-14 = y
Or, we can write it as:
y = -14
And that's it! We've solved for 'y'. The value of 'y' that makes this equation true is -14. Give yourselves a pat on the back – you've successfully navigated this algebraic challenge! This final step usually involves dividing or multiplying to isolate the variable completely. Remember, the goal is to get 'y' all by itself, so you know its value. Congratulations on making it to the end!
Conclusion
So, to recap, we solved the equation 7 = 7(5 + 3y) - 19y step by step. We distributed, combined like terms, isolated the 'y' term, and finally solved for 'y'. We found that y = -14. Solving equations like this is a fundamental skill in algebra, and you'll use it in many different areas of math and science.
Remember, the key is to break down the problem into smaller, manageable steps and to stay organized. Don't be afraid to take your time and double-check your work. With practice, you'll become a pro at solving for variables! I hope this step-by-step guide was helpful. Keep practicing, and you'll master these skills in no time. You guys rock!
If you ever get stuck on another equation, just remember these steps, and you'll be well on your way to finding the solution. Keep up the great work, and I'll catch you in the next math adventure!
