Solving 6x + 15 = 81: A Step-by-Step Guide
Hey guys! Today, we're diving into a fun math problem: solving the equation 6x + 15 = 81. It might look a little intimidating at first, but don't worry! We're going to break it down step by step so you can see how easy it actually is. We’ll not only find the value of x but also verify our solution to make sure we got it right. So, grab your pencils and let's get started!
Understanding the Basics of Algebraic Equations
Before we jump into solving the equation, let's quickly recap what an algebraic equation is all about. Think of an equation like a balanced scale. On one side, you have an expression (like 6x + 15), and on the other side, you have another expression or a value (in this case, 81). The equals sign (=) tells us that both sides are perfectly balanced. Our goal is to find the value of the unknown, which is x in this case, that keeps the scale balanced.
In the equation 6x + 15 = 81, x is a variable, meaning it represents a number we don't know yet. The number 6 in front of the x is called a coefficient. It tells us that we have 6 times the value of x. The number 15 is a constant, meaning its value doesn't change. The number 81 is also a constant. To solve for x, we need to isolate it on one side of the equation. This means getting x by itself, with no other numbers added, subtracted, multiplied, or divided with it. We do this by performing operations on both sides of the equation to maintain the balance. Remember, whatever you do to one side, you must do to the other!
Understanding these basic components is crucial for tackling algebraic equations. It's like knowing the ingredients of a recipe before you start cooking. So, now that we have a solid foundation, let’s move on to the actual steps of solving our equation. We’ll start by getting rid of that pesky constant term on the same side as x.
Step 1: Isolating the Term with 'x'
Our first mission is to isolate the term with x, which is 6x. Currently, we have 6x + 15 on the left side of the equation. To get 6x by itself, we need to eliminate the +15. How do we do that? We use the inverse operation. The inverse operation of addition is subtraction, so we'll subtract 15 from both sides of the equation. This is where the balance comes in – we have to do it to both sides to keep the equation true!
So, let's subtract 15 from both sides:
6x + 15 - 15 = 81 - 15
On the left side, +15 and -15 cancel each other out, leaving us with just 6x. On the right side, 81 - 15 equals 66. Our equation now looks like this:
6x = 66
Great! We've successfully isolated the term with x. Now, we're one step closer to finding the value of x. But we still have that coefficient 6 hanging around. It's multiplying x, so to get x completely alone, we need to use the inverse operation of multiplication. What is that, you ask? It's division! So, let's move on to the next step and divide both sides by 6.
Step 2: Solving for 'x'
Now that we have 6x = 66, it's time to get x all by itself. As we discussed, the 6 is multiplying x, so we need to do the opposite: divide. We're going to divide both sides of the equation by 6. Remember, whatever we do to one side, we have to do to the other to maintain that perfect balance!
So, let's divide both sides by 6:
(6x) / 6 = 66 / 6
On the left side, the 6 in the numerator and the 6 in the denominator cancel each other out, leaving us with just x. On the right side, 66 divided by 6 equals 11. Our equation now looks like this:
x = 11
Woohoo! We've done it! We've found the value of x. It turns out that x equals 11. But before we celebrate too much, we need to make sure we didn't make any mistakes along the way. That's where the next crucial step comes in: verifying our solution. We'll plug our value of x back into the original equation and see if it holds true. Let's jump into it!
Step 3: Verifying the Solution
We've solved for x, and we think it's 11. But in math, it's always a good idea to double-check your work. This is where verification comes in. To verify our solution, we're going to take the value we found for x (which is 11) and plug it back into the original equation: 6x + 15 = 81. If our solution is correct, then when we substitute 11 for x, the left side of the equation should equal the right side.
So, let's substitute x with 11:
6(11) + 15 = 81
Now, we need to simplify the left side of the equation. First, we multiply 6 by 11, which gives us 66:
66 + 15 = 81
Next, we add 66 and 15, which gives us 81:
81 = 81
Look at that! The left side of the equation equals the right side. This means our solution, x = 11, is correct! We've successfully solved the equation and verified our answer. Give yourselves a pat on the back! Verification is a super important step because it helps you catch any mistakes and ensures you have the correct solution. Now, let's wrap things up and summarize what we've learned.
Conclusion: Mastering Algebraic Equations
Alright, guys, we did it! We successfully solved the equation 6x + 15 = 81 and found that x equals 11. We also verified our solution to make sure we got it right. Along the way, we learned some valuable skills for tackling algebraic equations. We talked about the importance of understanding the basics, like what variables, coefficients, and constants are. We also learned about using inverse operations to isolate the variable and solve for its value. And, most importantly, we learned the importance of verifying our solution to ensure accuracy.
Remember, solving algebraic equations is like building with LEGOs. Each step is a brick, and when you put them together correctly, you get a beautiful structure (or, in this case, the correct solution!). So, don't be afraid to break down problems into smaller, manageable steps. Practice makes perfect, so the more equations you solve, the more confident you'll become. Keep up the great work, and I'll see you in the next math adventure!