Solving Linear Equations: -2.6b + 4 = 0.9b - 17
Hey guys! Today, we're diving into a fun little algebra problem. We've got the equation -2.6b + 4 = 0.9b - 17, and our mission, should we choose to accept it, is to find the value of 'b'. Don't worry; it's not as daunting as it looks! We'll break it down step by step, so even if algebra isn't your best buddy, you'll be able to follow along. Grab your pencils and let's get started!
Understanding the Equation
Before we jump into solving, let's take a moment to understand what we're looking at. We have a linear equation with one variable, 'b'. The goal is to isolate 'b' on one side of the equation so we can determine its value. The equation tells us that -2.6 times 'b', plus 4, is equal to 0.9 times 'b', minus 17. Our job is to find the value of 'b' that makes this statement true. Think of it like a puzzle where 'b' is the missing piece. To find it, we'll use some basic algebraic operations, like adding and subtracting terms from both sides of the equation, and then dividing to get 'b' all by itself. Remember, whatever we do to one side of the equation, we must do to the other to keep it balanced. This principle is the key to solving any algebraic equation, ensuring that the equality remains valid throughout our calculations. So, with our puzzle laid out before us, let's start moving the pieces around to reveal the value of 'b'.
Step 1: Combine Like Terms
The golden rule in algebra is to keep things organized. Our first step is to gather all the terms with 'b' on one side of the equation. To do this, we'll add 2.6b to both sides of the equation. This gets rid of the -2.6b on the left side and moves it over to the right side. So, our equation now looks like this: 4 = 0.9b + 2.6b - 17. Next, we simplify the right side by combining the 'b' terms. Adding 0.9b and 2.6b gives us 3.5b. Now our equation is even simpler: 4 = 3.5b - 17. See? We're making progress already! By strategically adding the same term to both sides, we've managed to consolidate the variable terms, bringing us one step closer to isolating 'b'. This process of combining like terms is a fundamental technique in algebra, making equations easier to work with and paving the way for solving them efficiently. So, let's keep this momentum going and move on to the next step, where we'll isolate the 'b' term even further.
Step 2: Isolate the Variable Term
Alright, now that we've got all the 'b' terms on one side, it's time to isolate that 3. 5b term. Currently, we have 4 = 3.5b - 17. To get the 3.5b by itself, we need to get rid of that -17. How do we do that? Easy! We add 17 to both sides of the equation. This cancels out the -17 on the right side and adds 17 to the left side. So, our equation transforms into 4 + 17 = 3.5b. Now, we simplify the left side by adding 4 and 17, which gives us 21. So, the equation now reads 21 = 3.5b. We're getting closer and closer! By adding 17 to both sides, we've successfully isolated the term containing our variable 'b'. This step is crucial because it sets us up to finally solve for 'b' in the next stage. Isolating the variable term is like clearing the path so we can zoom in on the variable itself. So, with the variable term now standing alone, let's proceed to the final step where we'll uncover the value of 'b'.
Step 3: Solve for 'b'
Here comes the grand finale! We've got 21 = 3.5b, and now we need to find out what 'b' equals. To do this, we'll divide both sides of the equation by 3.5. This will isolate 'b' on the right side and give us its value on the left side. So, we have 21 / 3.5 = b. When we do the division, we find that 21 divided by 3.5 is 6. Therefore, b = 6. Hooray! We've solved for 'b'! After all our hard work, we've discovered that the value of 'b' that satisfies the original equation is 6. Dividing both sides by the coefficient of 'b' is the final step in isolating the variable and determining its value. This process is like the last piece of the puzzle falling into place, revealing the solution we've been striving for. So, with 'b' now unveiled, let's take a moment to celebrate our algebraic victory!
Step 4: Verification
To make sure we didn't make any sneaky mistakes along the way, let's plug our solution, b = 6, back into the original equation: -2.6b + 4 = 0.9b - 17. Substituting b = 6, we get -2.6(6) + 4 = 0.9(6) - 17. Now, let's simplify each side. On the left side, -2.6 times 6 is -15.6, so we have -15.6 + 4. Adding those together gives us -11.6. On the right side, 0.9 times 6 is 5.4, so we have 5.4 - 17. Subtracting those gives us -11.6. So, we have -11.6 = -11.6. Yay! Both sides are equal, which means our solution, b = 6, is correct! Verification is an important step in problem-solving, as it confirms the accuracy of our solution and ensures that we haven't made any errors in our calculations. By substituting the value of 'b' back into the original equation, we've validated our answer and gained confidence in our result. So, with our solution verified, we can confidently say that b = 6 is indeed the correct answer.
Final Answer
So, after all that algebraic maneuvering, we've arrived at our final answer: b = 6. Nice job, team! Solving equations like this is a fundamental skill in algebra, and mastering it opens the door to tackling more complex problems. Remember, the key is to stay organized, keep the equation balanced, and take it one step at a time. Whether you're solving for 'b' or any other variable, the same principles apply. And with practice, you'll become an algebra whiz in no time! So, keep practicing, keep exploring, and keep having fun with math! You've got this! And remember, if you ever get stuck, don't hesitate to ask for help or review the steps we've covered here. Math is a journey, and every problem is an opportunity to learn and grow. So, keep pushing forward, and you'll achieve great things!
b = 6
