Match Equations To Solutions: An Easy Guide For Everyone

Welcome to the World of Equations: Why This Matters!

Hey there, math enthusiasts and curious minds! Ever looked at a string of numbers and letters, like something out of a secret code, and wondered, "What in the world am I supposed to do with this?" Well, today we're going to demystify those very things—what we call equations! Think of an equation as a perfectly balanced seesaw. Whatever you do to one side, you must do to the other to keep it level. Our mission today is to figure out the secret number, often represented by a letter like x, n, or b, that makes the seesaw perfectly balanced. It's like being a detective, searching for a hidden value!

Solving equations isn't just some abstract math concept confined to textbooks; it's a fundamental skill that pops up everywhere, often without you even realizing it. From figuring out how much change you'll get back at the store to calculating the perfect recipe adjustments for a big party, or even more complex tasks like budgeting your finances or understanding scientific formulas, the ability to manipulate and solve equations is super powerful. It helps you think logically, problem-solve efficiently, and understand the relationships between different quantities. Many people find algebra intimidating, but honestly, with a little guidance and a friendly approach, you'll see that it's just a set of logical steps. We’ll break down each problem, explaining the why behind every what, making sure you're not just memorizing steps, but truly understanding the journey from a complex equation to its simple solution. Get ready to boost your confidence and unlock a whole new level of mathematical prowess. We're talking about mastering the art of finding those elusive numbers that make everything click into place. So, buckle up, grab a pen and paper if you like, and let's dive into this exciting adventure of matching equations with their solutions together! We'll make it fun, straightforward, and totally achievable, because you absolutely can become an equation-solving pro!

Your Toolkit for Tackling Equations: Key Concepts

Before we jump into our specific problems, let's quickly review some essential tools and concepts that will make solving equations a breeze. Think of these as your trusty gadgets in your mathematical toolkit. First up, we have variables. These are the letters (like x, n, b) that represent an unknown number. Our main goal is almost always to isolate this variable, meaning we want to get it all by itself on one side of the equation. Next, we have constants, which are just fixed numbers—they don't change. Then, there's the concept of inverse operations. This is super crucial! Every mathematical operation has an opposite: addition undoes subtraction, and vice-versa; multiplication undoes division, and vice-versa. To move a number from one side of the equation to the other, you always perform the inverse operation. For example, if you see +5 on one side, you'll subtract 5 from both sides to cancel it out. If you see *3, you'll divide by 3 on both sides. Remember that seesaw analogy? Whatever you do to one side, you must do to the other to maintain balance.

Another vital concept is the distributive property. This comes into play when you have a number outside parentheses, like 2(4 + 7b). It means you multiply the number outside by each term inside the parentheses. So, 2(4 + 7b) becomes (2 * 4) + (2 * 7b), which simplifies to 8 + 14b. Without correctly applying this property, your equation will quickly go off track! Finally, don't forget combining like terms. If you have n - 6 + 8n, you can combine n and 8n to get 9n. You can only combine terms that have the exact same variable raised to the exact same power. So, you can add 2x and 3x to get 5x, but you can't combine 2x and 3x² or 2x and 3y. Keeping these fundamental ideas in mind will equip you with everything you need to confidently approach any linear equation. We’ll be applying these concepts directly in the problems ahead, so seeing them in action will solidify your understanding. Get ready to apply these brilliant tools and become an equation-solving wizard!

Let's Solve Together: Equation by Equation

Alright, guys, it's showtime! We've got our tools, we understand the basics, and now it's time to roll up our sleeves and tackle these equations one by one. Our goal is to solve each equation and then match its solution to the correct option from our list: a. 8, b. 0, c. 6, d. 3. Let's dive in and see how we can uncover those hidden values!

Equation 1: Solving for 'x' in $-3=1-x+2$

This first equation might look a little messy with numbers and a negative x, but trust me, it’s straightforward once we break it down. Our equation is: $-3 = 1 - x + 2$. The ultimate goal, remember, is to get x all by itself. Always start by simplifying each side of the equation as much as possible before moving terms across the equals sign. On the right side, we have constants 1 and 2 that can be combined. So, 1 + 2 gives us 3.

So, the equation simplifies to: $-3 = 3 - x$.

Now, we need to get x alone. Currently, we have 3 on the same side as -x. To move that 3 to the other side, we need to perform the inverse operation. Since it's a positive 3 (implied +3), we'll subtract 3 from both sides of the equation. This maintains the balance, just like our seesaw!

$-3 - 3 = 3 - x - 3$

On the left side, -3 - 3 combines to -6. On the right side, 3 - 3 cancels out, leaving us with just -x. So now we have:

$-6 = -x$

We're really close! We have -x, but we want positive x. This means x is being multiplied by -1 (because -x is the same as -1 * x). To get rid of that -1, we'll perform the inverse operation: divide both sides by -1.

$-6 / -1 = -x / -1$

And voilà! A negative divided by a negative equals a positive. So, on the left, -6 / -1 becomes 6. On the right, -x / -1 becomes x.

$6 = x$

So, the solution for the first equation is x = 6. Now, let’s check our options: a. 8, b. 0, c. 6, d. 3. Our answer, 6, perfectly matches option c. See? Not so bad at all! This problem really highlights the importance of simplifying first and being careful with negative signs when combining or dividing. It’s a common spot for little errors, so taking your time here pays off big time in getting to the correct solution. Always double-check your arithmetic, especially when dealing with integers, to ensure accuracy throughout the entire process. You've got this! This foundational problem sets a great precedent for the more complex ones to come, building a strong understanding of isolating variables step by simple step. Keep that confidence high!

Equation 2: Unraveling 'n' in $n-6+8 n=-6$

Alright, moving on to our second challenge: $n - 6 + 8n = -6$. This one introduces multiple terms with the same variable, n. As we discussed in our toolkit section, the very first step when you see this is to combine like terms on each side of the equation. On the left side, we have n and +8n. Remember, a lone n is the same as 1n. So, combining 1n + 8n gives us 9n. The -6 is a constant and doesn't have a variable n attached to it, so it stays as it is for now.

After combining like terms, our equation looks much neater: $9n - 6 = -6$.

Now, our goal is to isolate n. The 9n term is currently being modified by a -6. To undo this subtraction, we'll perform the inverse operation, which is addition. So, we're going to add 6 to both sides of the equation to keep it balanced.

$9n - 6 + 6 = -6 + 6$

On the left side, -6 + 6 cancels out, leaving us with just 9n. On the right side, -6 + 6 also equals 0. This is a really important result because it tells us something specific about n!

So, the equation now is: $9n = 0$.

We're almost there! n is currently being multiplied by 9. To get n by itself, we need to perform the inverse operation of multiplication, which is division. We'll divide both sides of the equation by 9.

$9n / 9 = 0 / 9$

On the left side, 9n / 9 simplifies to n. And on the right side, 0 / 9 is simply 0. Any time you divide zero by any non-zero number, the result is always zero. Don't let that 0 throw you off; it's a perfectly valid and common solution in algebra!

$n = 0$

So, the solution for the second equation is n = 0. Let's check our options: a. 8, b. 0, c. 6, d. 3. Our answer, 0, matches option b. Fantastic work! This problem truly emphasizes the power of combining like terms early on to simplify the equation, making the subsequent steps much clearer and less prone to errors. It also serves as a great reminder that zero is a perfectly legitimate and frequently encountered solution in algebra. Understanding why zero comes up helps solidify your overall grasp of equation solving, showing that sometimes, the simplest answer is the correct one. Keep your problem-solving momentum going; you're doing awesome!

Equation 3: Cracking the Code for 'b' in $2(4+7 b)-3=117$

Alright, time for a slightly more involved one, but nothing we can't handle! Our third equation is $2(4+7 b)-3=117$. See those parentheses? That's our cue to use the distributive property first. The number 2 outside the parentheses needs to be multiplied by each term inside the parentheses. So, 2 * 4 gives us 8, and 2 * 7b gives us 14b.

Applying the distributive property transforms the equation into: $8 + 14b - 3 = 117$.

Now, before we start moving terms across the equals sign, let's simplify the left side by combining like terms. We have two constant terms on the left: 8 and -3. Combining these, 8 - 3, gives us 5. The 14b term doesn't have any other b terms to combine with, so it stays as 14b.

The simplified equation now looks like this: $14b + 5 = 117$.

Much cleaner, right? Now, our goal is to isolate b. The 14b term is currently joined by a +5. To get rid of this +5, we perform the inverse operation: subtract 5 from both sides of the equation.

$14b + 5 - 5 = 117 - 5$

On the left, +5 - 5 cancels out, leaving 14b. On the right, 117 - 5 equals 112.

So now we have: $14b = 112$.

We're almost there! b is being multiplied by 14. To finally get b by itself, we'll perform the inverse operation: divide both sides of the equation by 14.

$14b / 14 = 112 / 14$

On the left, 14b / 14 simplifies to b. And on the right, 112 / 14 works out to 8. You might need to do a quick mental calculation or a bit of long division here, but it definitely comes out to a nice whole number!

$b = 8$

And there we have it! The solution for the third equation is b = 8. Let's check our options: a. 8, b. 0, c. 6, d. 3. Our answer, 8, is a perfect match for option a. This problem was a fantastic demonstration of how to handle parentheses using the distributive property and then proceed with combining terms and inverse operations. It's a classic multi-step equation, and mastering it means you're really getting the hang of this whole equation-solving thing. The methodical approach ensures that even complex expressions become manageable, one step at a time. Great job tackling this one, team! You're building solid mathematical muscles here, and each successful solve reinforces your understanding. Keep pushing through!

Equation 4: Decoding 'x' in $4(-7-8 x)+7=-117$

Last but not least, let's tackle our final equation: $4(-7-8 x)+7=-117$. Just like the previous problem, those parentheses are a big hint that our first move needs to be the distributive property. We'll multiply the 4 outside by each term inside the parentheses. Be extra careful with the negative signs here, guys!

So, 4 * -7 gives us -28. And 4 * -8x gives us -32x.

Applying the distributive property, our equation becomes: $-28 - 32x + 7 = -117$.

Now, before we start shifting numbers around, let's combine like terms on the left side. We have two constant terms: -28 and +7. Combining these, -28 + 7 equals -21. The -32x term doesn't have any other x terms to combine with, so it stays as -32x.

The simplified equation is: $-21 - 32x = -117$.

Our mission is to isolate x. The term -32x is currently accompanied by a -21. To move this -21 to the other side, we perform the inverse operation: add 21 to both sides of the equation.

$-21 - 32x + 21 = -117 + 21$

On the left, -21 + 21 cancels out, leaving us with just -32x. On the right, -117 + 21 gives us -96. Remember, when adding numbers with different signs, you essentially subtract their absolute values and keep the sign of the larger number.

So now we have: $-32x = -96$.

We're in the final stretch! x is being multiplied by -32. To get x by itself, we'll perform the inverse operation: divide both sides by -32.

$-32x / -32 = -96 / -32$

On the left, -32x / -32 simplifies to x. On the right, -96 / -32. A negative divided by a negative results in a positive. And 96 divided by 32 equals 3.

$x = 3$

Fantastic! The solution for the fourth equation is x = 3. Checking our options: a. 8, b. 0, c. 6, d. 3. Our answer, 3, matches option d. This problem was a great test of careful distribution, especially with negative numbers, and meticulous combining of like terms. It also reinforces the rule of dividing negatives to get a positive. If you successfully navigated this one, give yourself a huge pat on the back – you're really getting the hang of solving complex equations! Each equation has presented a slightly different twist, and your ability to adapt and apply the correct principles demonstrates a solid understanding. Keep that momentum going, because you’re well on your way to becoming an equation master!

Quick Match-Up: Connecting Solutions to Equations

Now that we've gone through each equation step-by-step and found their solutions, let's quickly put them all together and officially match them up to the given options:

• Equation 1: $-3=1-x+2$

  • Solution: $x = 6$
  • Matches: c. 6

• Equation 2: $n-6+8 n=-6$

  • Solution: $n = 0$
  • Matches: b. 0

• Equation 3: $2(4+7 b)-3=117$

  • Solution: $b = 8$
  • Matches: a. 8

• Equation 4: $4(-7-8 x)+7=-117$

  • Solution: $x = 3$
  • Matches: d. 3

There you have it! A perfect match for every single equation. This systematic approach ensures accuracy and builds confidence in your problem-solving abilities.

Beyond the Basics: Tips for Becoming an Equation Master

Solving these specific problems is a fantastic start, but becoming a true equation master is an ongoing journey, and I've got a few more tips to help you on your way. First, and perhaps most importantly, practice, practice, practice! The more equations you solve, the more familiar you'll become with different types of problems and the quicker you'll spot the necessary steps. Don't shy away from challenging yourself with new variations. Second, always check your answers. Once you find a solution, plug it back into the original equation. If both sides of the equation are equal after you substitute your value, then you know your solution is correct! This simple step can save you from making errors and is a powerful self-correction tool. Third, stay organized. Write out each step clearly. Don't try to do too much in your head, especially with longer problems. A neat, step-by-step approach not only helps you avoid mistakes but also makes it easier to review your work if you get stuck. Finally, don't be afraid to make mistakes! Mistakes are learning opportunities. Figure out where you went wrong, understand why it was wrong, and apply that lesson moving forward. Every error brings you closer to mastery. Embrace the challenge, and you'll soon find yourself tackling even the trickiest equations with ease and confidence. Remember, mathematics is a skill, and like any skill, it improves with consistent effort and a positive attitude.

Wrapping It Up: You're an Equation Whiz!

Wow, you've made it through! From understanding the fundamental principles of solving equations to meticulously breaking down and finding the solution for each problem, you've done an amazing job. We’ve covered combining like terms, mastering the distributive property, and applying inverse operations to isolate variables—all crucial skills for any budding mathematician. You've not just memorized answers; you've understood the process, and that's what truly matters. Remember, the journey to becoming proficient in algebra is all about patience, practice, and a willingness to learn from every problem. Keep honing those skills, keep asking questions, and never stop being curious. With the foundational knowledge you’ve gained today, you're now well-equipped to tackle a wide array of algebraic challenges. You are officially an equation-solving whiz! Keep up the fantastic work, and keep exploring the wonderful world of mathematics!