Finding Equations Equivalent To 7x + 20 = 6 A Step By Step Guide

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Hey guys! Ever stumbled upon an equation and felt like you're in a maze? Don't worry, we've all been there. Today, we're going to break down a common type of math problem: finding equations that are equivalent to a given one. Specifically, we'll be tackling the equation 7x + 20 = 6. Our mission? To figure out which of the following equations is just another way of saying the same thing:

  • 3x - 5 = 11
  • 7x - 12 = 2
  • 10 - 3x = 16
  • 15 + 2x = 19

Let's dive in and make sense of this together!

Understanding Equivalent Equations

First things first, what does it even mean for equations to be equivalent? Think of it like different languages saying the same sentence. Equivalent equations are equations that have the exact same solution. In simpler terms, if you solve them, you'll get the same value for 'x'. This concept is super important in algebra because it allows us to manipulate equations into simpler forms without changing their core meaning. This simplification is what we will be focusing on today. To identify equivalent equations, our main goal is to isolate 'x' on one side of the equation. This usually involves performing inverse operations, such as adding or subtracting the same number from both sides or multiplying or dividing both sides by the same non-zero number. This process is crucial for maintaining the balance of the equation, ensuring that the left-hand side always equals the right-hand side. Once we've isolated 'x', we have the solution to the equation. We can then compare this solution to the solutions of other equations to determine if they are equivalent. Remember, equivalent equations are just different ways of expressing the same mathematical relationship, and understanding how to find them is a fundamental skill in algebra and beyond. So, let's embark on this journey of unraveling equations and mastering the art of identifying their equivalents!

Solving the Original Equation: 7x + 20 = 6

Before we go hunting for equivalent equations, we need to know what we're looking for! Let's solve the original equation, 7x + 20 = 6. This will give us the solution for 'x' that any equivalent equation must also have. This initial step is crucial, guys. It's like having the key to unlock all the other puzzles. To solve the equation 7x + 20 = 6, our primary goal is to isolate 'x' on one side. This involves a series of algebraic manipulations that carefully peel away the layers surrounding 'x'. We start by addressing the constant term that's added to '7x', which is +20. To counteract this addition, we perform the inverse operation: subtraction. We subtract 20 from both sides of the equation. Remember, guys, whatever you do to one side, you must do to the other to maintain the balance and integrity of the equation. So, subtracting 20 from both sides gives us: 7x + 20 - 20 = 6 - 20. Simplifying this, we get 7x = -14. Now we're one step closer! 'x' is currently being multiplied by 7. To isolate 'x' completely, we need to undo this multiplication. The inverse operation of multiplication is division. Therefore, we divide both sides of the equation by 7. This yields 7x / 7 = -14 / 7. Performing the division, we find that x = -2. Voila! We've solved the equation. The solution to 7x + 20 = 6 is x = -2. This means that any equation equivalent to the original equation must also have x = -2 as its solution. Keep this value locked in your memory, guys, it's our golden ticket for finding the equivalent equation among the options. Now that we have this crucial piece of information, we can proceed to examine the other equations and see which one shares the same solution.

Checking the First Option: 3x - 5 = 11

Alright, let's put on our detective hats and investigate the first potential equivalent equation: 3x - 5 = 11. To determine if this equation is indeed equivalent to our original equation, 7x + 20 = 6, we need to solve it for 'x' and see if we arrive at the same solution we found earlier, which was x = -2. Remember, guys, this is the golden rule: equivalent equations have the same solution. Let's dive into the process of solving 3x - 5 = 11. Our mission, as always, is to isolate 'x' on one side of the equation. We start by looking at the constant term that's affecting 'x'. In this case, we have -5. To counteract this subtraction, we need to perform the inverse operation: addition. We add 5 to both sides of the equation. Adding 5 to both sides maintains the balance and ensures we're not changing the fundamental truth of the equation. This gives us: 3x - 5 + 5 = 11 + 5. Simplifying this, we get 3x = 16. We're getting closer! 'x' is now being multiplied by 3. To isolate 'x' completely, we need to undo this multiplication. The inverse operation of multiplication is division. Therefore, we divide both sides of the equation by 3. This yields 3x / 3 = 16 / 3. Performing the division, we find that x = 16/3, which is approximately 5.33. Hmmm... This is interesting. We solved the equation 3x - 5 = 11 and found that x = 16/3. However, our original equation, 7x + 20 = 6, had a solution of x = -2. These solutions are different. This means that the equation 3x - 5 = 11 is not equivalent to 7x + 20 = 6. It's like they're speaking different mathematical languages. So, we can confidently cross this option off our list and move on to the next potential equivalent equation.

Examining the Second Option: 7x - 12 = 2

Okay, guys, let's move on to our second suspect: the equation 7x - 12 = 2. Our mission remains the same: to solve this equation for 'x' and see if it matches the solution of our original equation, 7x + 20 = 6, which we know is x = -2. Remember, we're on the hunt for equivalent equations, which means they must have the same solution. To solve 7x - 12 = 2, we follow our trusty algebraic steps. Our goal is to isolate 'x' on one side of the equation. First, we tackle the constant term that's affecting 'x'. We have -12. To undo this subtraction, we perform the inverse operation: addition. We add 12 to both sides of the equation. Adding the same value to both sides keeps the equation balanced and true. This gives us: 7x - 12 + 12 = 2 + 12. Simplifying this, we get 7x = 14. Great! We're one step closer to isolating 'x'. 'x' is currently being multiplied by 7. To completely isolate 'x', we need to undo this multiplication. The inverse operation of multiplication is division. Therefore, we divide both sides of the equation by 7. This yields 7x / 7 = 14 / 7. Performing the division, we find that x = 2. Now, let's compare this result to the solution of our original equation. We found that 7x - 12 = 2 has a solution of x = 2. But, our original equation, 7x + 20 = 6, has a solution of x = -2. These solutions are not the same. This tells us that the equation 7x - 12 = 2 is not equivalent to 7x + 20 = 6. They have different solutions, so they can't be saying the same thing. We're making progress in our investigation, though! We've eliminated another option, bringing us closer to finding the true equivalent equation.

Analyzing the Third Equation: 10 - 3x = 16

Alright detectives, let's investigate the third suspect in our lineup: the equation 10 - 3x = 16. Remember, we're on a quest to find an equation that's equivalent to 7x + 20 = 6. This means the solution for 'x' must be the same as what we found earlier, which is x = -2. To solve 10 - 3x = 16, our goal is, as always, to isolate 'x'. But this one looks a little different, right? The 'x' term is negative. Don't sweat it, guys! We'll tackle it step by step. First, let's deal with the constant term on the side with 'x'. We have 10, which is the same as +10. To get rid of it, we perform the inverse operation: subtraction. We subtract 10 from both sides of the equation. Subtracting the same value from both sides keeps the equation balanced. This gives us: 10 - 3x - 10 = 16 - 10. Simplifying this, we get -3x = 6. Now we're getting closer, but we still have that pesky negative sign in front of the '3x'. Remember, -3x is the same as -3 multiplied by x. To isolate 'x', we need to undo this multiplication. The inverse operation of multiplication is division. So, we divide both sides of the equation by -3. It's crucial to divide by the entire coefficient, including the negative sign, to ensure 'x' becomes positive. This yields -3x / -3 = 6 / -3. Performing the division, we find that x = -2. Eureka! Let's take a moment to celebrate this mathematical breakthrough. We solved the equation 10 - 3x = 16 and found that x = -2. Now, let's compare this to the solution of our original equation. Our original equation, 7x + 20 = 6, also has a solution of x = -2. Boom! We have a match! The equation 10 - 3x = 16 is equivalent to 7x + 20 = 6. They speak the same mathematical language. We've found our equivalent equation!

Verifying the Fourth Option: 15 + 2x = 19

Just to be absolutely sure, guys, and to show how the process works, let's quickly check the last option: 15 + 2x = 19. Even though we've already found an equivalent equation, it's always good practice to verify and solidify our understanding. We're looking to see if this equation also has the solution x = -2, which is the solution to our original equation, 7x + 20 = 6. To solve 15 + 2x = 19, we follow our familiar steps. We want to isolate 'x'. First, we deal with the constant term, which is +15. To undo this addition, we subtract 15 from both sides of the equation. This gives us: 15 + 2x - 15 = 19 - 15. Simplifying, we get 2x = 4. Now, 'x' is being multiplied by 2. To isolate 'x', we divide both sides of the equation by 2. This yields 2x / 2 = 4 / 2. Performing the division, we find that x = 2. Okay, let's compare this to our target solution. The equation 15 + 2x = 19 has a solution of x = 2. Our original equation, 7x + 20 = 6, has a solution of x = -2. These solutions are different. Therefore, the equation 15 + 2x = 19 is not equivalent to 7x + 20 = 6. It's always good to double-check, right? This confirms that our earlier finding was correct: 10 - 3x = 16 is the equivalent equation.

Conclusion: The Equivalent Equation

Alright, guys! We've done it! We've successfully navigated the world of equivalent equations. We started with the equation 7x + 20 = 6 and set out to find which of the given options was just another way of saying the same thing. By systematically solving each equation and comparing their solutions, we discovered that the equation 10 - 3x = 16 is the equivalent equation. It shares the same solution, x = -2, as our original equation. Remember, the key to finding equivalent equations is to solve for 'x' and compare the results. This process involves using inverse operations to isolate 'x' on one side of the equation, always maintaining balance by performing the same operation on both sides. We've seen how this works step-by-step, so you're now equipped to tackle similar problems with confidence. Keep practicing, and you'll become a master of equations in no time! You've got this!