Solving Equations Step By Step A Comprehensive Guide

Hey guys! Are you wrestling with equations? Don't sweat it! Solving equations can seem daunting, but with a systematic approach, you'll be cracking them like a pro in no time. This guide will walk you through simplifying and solving different types of equations, step by step. We'll break down each problem, making it super easy to follow along. So, grab your pencils, and let's dive in!

Understanding the Basics of Solving Equations

Before we jump into specific equations, let’s nail down the core principles. The main goal in solving an equation is to isolate the variable. Think of it like finding the missing piece of a puzzle. The variable, often represented by letters like 'x', 'y', 'h', or 'b', is the unknown value we need to figure out. To isolate it, we use inverse operations – operations that undo each other. Addition and subtraction are inverse operations, as are multiplication and division. Remember, whatever you do to one side of the equation, you must do to the other side to keep the equation balanced. This is the golden rule of equation solving!

Why is balancing equations so crucial? Imagine a seesaw. If you add weight to one side without adding the same weight to the other, the seesaw tips. Equations are the same way. The equals sign (=) represents that balance. To maintain the balance and find the correct value for our variable, we have to apply the same operations to both sides. This might seem a bit abstract now, but as we work through examples, you'll see exactly how it works in practice. Trust me, once you get this concept, solving equations becomes a whole lot easier. We're not just manipulating numbers; we're maintaining a delicate equilibrium!

Simplifying equations is another key step in the solving process. This involves combining like terms – terms that have the same variable raised to the same power (or are just constants). For instance, in the expression 9h + 8 - 12, the constants 8 and -12 can be combined. Simplifying makes the equation less cluttered and easier to work with. It's like decluttering your workspace before starting a big project; it helps you focus on what's important. So, before you start isolating the variable, always look for opportunities to simplify. This will save you time and reduce the chances of making mistakes. We'll see this in action as we tackle the problems below. Let's get started!

Solving Equation A: 9h + 8 - 12 = -40

Let's kick things off with our first equation: 9h + 8 - 12 = -40. Remember our goal? To isolate the variable 'h'. Before we start moving things around, let's simplify the equation. We see two constants on the left side: 8 and -12. We can combine these! What is 8 - 12? It's -4. So, we can rewrite our equation as 9h - 4 = -40. See how much cleaner that looks already?

Now, let’s isolate the term with 'h'. We have 9h - 4 = -40. We want to get the '9h' by itself. What's being done to '9h'? We're subtracting 4. To undo subtraction, we add! So, we'll add 4 to both sides of the equation. This gives us 9h - 4 + 4 = -40 + 4. The -4 and +4 on the left side cancel each other out, leaving us with 9h = -36. We're getting closer!

We're almost there! We have 9h = -36. Now, 'h' is being multiplied by 9. To undo multiplication, we divide. We'll divide both sides of the equation by 9. This gives us (9h) / 9 = (-36) / 9. The 9s on the left side cancel out, leaving us with just 'h'. On the right side, -36 divided by 9 is -4. So, we have h = -4. Boom! We've solved for 'h'! Remember to always double-check your answer by plugging it back into the original equation to make sure it holds true. This is like the final proofread on your work.

Solving Equation B: 9 + 6g + 1 = 100

Next up, we have equation B: 9 + 6g + 1 = 100. Just like before, our mission is to isolate the variable, which in this case is 'g'. And what's our first step? You guessed it – simplification! On the left side, we have two constants: 9 and 1. Let's combine them. 9 + 1 = 10. So, we can rewrite our equation as 10 + 6g = 100. Much better, right?

Now, let's get that '6g' term by itself. We have 10 + 6g = 100. What's being added to '6g'? The number 10 is. To undo addition, we subtract. So, we'll subtract 10 from both sides of the equation. This gives us 10 + 6g - 10 = 100 - 10. The 10 and -10 on the left side cancel each other out, leaving us with 6g = 90. We're on a roll!

We're in the home stretch! We have 6g = 90. Now, 'g' is being multiplied by 6. To undo multiplication, we divide. We'll divide both sides of the equation by 6. This gives us (6g) / 6 = 90 / 6. The 6s on the left side cancel out, leaving us with just 'g'. On the right side, 90 divided by 6 is 15. So, we have g = 15. Fantastic! We've solved for 'g'! And remember, always, always, always double-check your answer by plugging it back into the original equation. This small step can save you from making simple mistakes.

Solving Equation C: -45 = 28 - 13 + 10b

Alright, let's tackle equation C: -45 = 28 - 13 + 10b. By now, you know the drill! Our goal is to isolate the variable, 'b' in this case, and our first move is to simplify. Look at the right side of the equation. We have two constants: 28 and -13. Let’s combine them. What's 28 - 13? It's 15. So, we can rewrite our equation as -45 = 15 + 10b. Looking cleaner and clearer already!

Now, let's isolate the term with 'b'. We have -45 = 15 + 10b. What's being added to '10b'? The number 15 is. To undo addition, we subtract. So, we'll subtract 15 from both sides of the equation. This gives us -45 - 15 = 15 + 10b - 15. On the left side, -45 - 15 is -60. On the right side, the 15 and -15 cancel each other out, leaving us with -60 = 10b.

We're just one step away! We have -60 = 10b. Now, 'b' is being multiplied by 10. To undo multiplication, we divide. We'll divide both sides of the equation by 10. This gives us -60 / 10 = (10b) / 10. On the left side, -60 divided by 10 is -6. On the right side, the 10s cancel out, leaving us with just 'b'. So, we have -6 = b or, if you prefer, b = -6. Excellent! We've solved for 'b'! And, you know what comes next – the all-important double-check. Plug -6 back into the original equation to make sure everything balances out.

Solving Equation D: 37 = 11 + 3b - 19

Last but not least, let's conquer equation D: 37 = 11 + 3b - 19. You've become equation-solving experts by now, so you know the first step is always to simplify. Looking at the right side of the equation, we see two constants: 11 and -19. Let's combine them. What's 11 - 19? It's -8. So, we can rewrite our equation as 37 = -8 + 3b. Simplifying makes a world of difference!

Now, let's isolate the '3b' term. We have 37 = -8 + 3b. What's being added to '3b'? Actually, we're adding -8, which is the same as subtracting 8. To undo subtraction, we add! So, we'll add 8 to both sides of the equation. This gives us 37 + 8 = -8 + 3b + 8. On the left side, 37 + 8 is 45. On the right side, the -8 and +8 cancel each other out, leaving us with 45 = 3b.

We're down to the final step! We have 45 = 3b. Now, 'b' is being multiplied by 3. To undo multiplication, we divide. We'll divide both sides of the equation by 3. This gives us 45 / 3 = (3b) / 3. On the left side, 45 divided by 3 is 15. On the right side, the 3s cancel out, leaving us with just 'b'. So, we have 15 = b or, if you like it the other way, b = 15. Hooray! We've solved for 'b'! And, of course, the final step is to substitute 15 back into the original equation to verify that it works. Always that final check!

Conclusion: You're an Equation-Solving Pro!

And there you have it! We've walked through simplifying and solving four different equations. Remember the key takeaways: simplify first, use inverse operations to isolate the variable, and always check your answers. Solving equations is like building a muscle – the more you practice, the stronger you get. Don't be afraid to make mistakes; they're part of the learning process. Keep practicing, and you'll become an equation-solving master in no time. You've got this!