Solve For V The Ultimate Guide To Mastering Linear Equations

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Hey guys! 👋 Are you struggling with solving equations? Don't worry, we've all been there! Today, we're going to break down a common type of equation and show you exactly how to solve it. We'll tackle the equation -2v + 5(v - 5) = 2 step-by-step, making sure you understand the logic behind each move. So, grab your pencils and let's dive in!

Understanding the Equation: -2v + 5(v - 5) = 2

Before we jump into solving, let's take a closer look at our equation: -2v + 5(v - 5) = 2. This is a linear equation, which means it involves a variable (v in this case) raised to the power of 1. Our goal is to isolate the variable v on one side of the equation so we can determine its value. To do this, we'll use a combination of algebraic operations, making sure to maintain the balance of the equation (what we do to one side, we must do to the other). The heart of solving any equation lies in the application of fundamental algebraic principles, such as the distributive property and the combination of like terms. These principles act as the bedrock upon which we can methodically unravel the equation, ultimately revealing the value of the variable we seek. At the very core of algebra is the principle of maintaining balance. Think of an equation as a perfectly balanced scale. Any operation performed on one side must be mirrored on the other to preserve this balance. This ensures that the relationship between the two sides remains consistent throughout the solving process. In the context of our equation, this means that every addition, subtraction, multiplication, or division we perform on the left side must be identically applied to the right side. This fundamental concept underpins the entire process of equation solving and is crucial for arriving at the correct solution. By adhering to this principle, we can confidently manipulate the equation while ensuring that the equality remains intact, guiding us towards the value of the variable we are solving for. The equation -2v + 5(v - 5) = 2 presents a perfect opportunity to illustrate this principle in action. As we work through the steps, you'll see how carefully we apply each operation to both sides, ensuring that the balance is maintained and the integrity of the equation is preserved. This methodical approach not only helps us find the correct solution but also reinforces the core principles of algebraic manipulation. So, let's roll up our sleeves and delve into the steps, always keeping in mind the golden rule of balance in algebra!

Step 1: Distribute the 5

Okay, first things first, we need to get rid of those parentheses! To do this, we'll use the distributive property. This means we multiply the 5 outside the parentheses by each term inside the parentheses: 5 * v and 5 * -5. Applying the distributive property is like unwrapping a present – it reveals the individual components hidden inside. In our equation, the parentheses act as a wrapper, bundling the terms (v - 5) together. By multiplying the 5 by each term inside, we're essentially unwrapping this bundle and making the individual components visible. This step is crucial because it allows us to simplify the equation and move closer to isolating the variable. Think of it as preparing the ingredients before you start cooking – you need to unpack and organize everything before you can begin the main task. In the same way, distributing the 5 sets the stage for the next steps in solving the equation. It lays the groundwork for combining like terms and ultimately isolating the variable. Now, let's see the distributive property in action. Multiplying 5 by v gives us 5v, and multiplying 5 by -5 gives us -25. So, the equation -2v + 5(v - 5) = 2 transforms into -2v + 5v - 25 = 2. See how the parentheses have disappeared, and we now have a more straightforward equation to work with? This transformation is the magic of the distributive property at play. It's a powerful tool that simplifies complex expressions and paves the way for solving equations. So, remember, whenever you see parentheses in an equation, the distributive property is your best friend. It's the key to unlocking the hidden potential of the expression and moving closer to the solution. With this step under our belt, we're well on our way to conquering the equation and finding the value of v. So, let's move on to the next step and continue our journey towards the solution!

This gives us:

-2v + 5v - 25 = 2

Step 2: Combine Like Terms

Now that we've distributed the 5, we have some like terms on the left side of the equation. Like terms are terms that have the same variable raised to the same power. In our case, we have -2v and 5v. We can combine these terms by simply adding their coefficients (the numbers in front of the variable). Combining like terms is like sorting your socks – you group the ones that are similar together to make things simpler. In our equation, -2v and 5v are like terms because they both have the variable 'v' raised to the power of 1. We can combine them just like we would combine two socks of the same color. Think of it as simplifying a recipe – you combine similar ingredients to create a more cohesive dish. In the same way, combining like terms makes our equation more manageable and easier to solve. It's a fundamental step in the equation-solving process that helps us streamline the expression and get closer to isolating the variable. Now, let's see how we combine -2v and 5v. We simply add their coefficients: -2 + 5 = 3. So, -2v + 5v becomes 3v. This is like having 5 apples and taking away 2 – you're left with 3 apples. By combining these like terms, we've reduced the number of terms in our equation and made it more concise. This simplification is crucial because it brings us one step closer to isolating the variable 'v'. Imagine trying to navigate a cluttered room – it's much easier to move around once you've organized things. In the same way, combining like terms declutters our equation and makes it easier to solve. So, remember, whenever you see like terms in an equation, don't hesitate to combine them. It's a powerful tool that simplifies the expression and paves the way for finding the solution. With this step completed, we're well on our way to solving the equation. So, let's move on to the next step and continue our journey!

Combining -2v and 5v, we get 3v. Our equation now looks like this:

3v - 25 = 2

Step 3: Isolate the Variable Term

Our next goal is to isolate the term with the variable (3v) on one side of the equation. To do this, we need to get rid of the -25 on the left side. We can do this by adding 25 to both sides of the equation. Remember, whatever we do to one side, we must do to the other to keep the equation balanced! Isolating the variable term is like clearing a path to your destination – you remove any obstacles that are in the way. In our equation, -25 is an obstacle that's preventing us from getting '3v' alone on one side. To remove this obstacle, we perform the opposite operation: adding 25 to both sides. Think of it as balancing a seesaw – if one side is too heavy, you add weight to the other side to level it out. In the same way, adding 25 to both sides of the equation maintains the balance and allows us to isolate the variable term. This step is crucial because it brings us closer to the ultimate goal of solving for 'v'. By isolating the variable term, we're essentially setting up the final step: dividing both sides by the coefficient of 'v'. Now, let's see how adding 25 to both sides works. On the left side, -25 + 25 cancels out, leaving us with just 3v. On the right side, 2 + 25 gives us 27. So, the equation 3v - 25 = 2 transforms into 3v = 27. See how we've successfully isolated the variable term? This transformation is a key step in solving the equation. It's like building a bridge – we've created a clear path from the variable term to the solution. So, remember, whenever you need to isolate a term, perform the opposite operation on both sides of the equation. It's a powerful technique that helps you move closer to the solution. With this step completed, we're almost there! Let's move on to the final step and conquer this equation!

Adding 25 to both sides:

3v - 25 + 25 = 2 + 25

This simplifies to:

3v = 27

Step 4: Solve for v

We're almost there! Now we have 3v = 27. To solve for v, we need to get v by itself. Since v is being multiplied by 3, we'll do the opposite operation: divide both sides of the equation by 3. Solving for the variable is like unlocking a treasure chest – you're finally getting to the prize you've been working towards. In our equation, the variable 'v' is the treasure, and we've been using algebraic tools to unlock it. To get 'v' by itself, we need to undo the multiplication by 3. We do this by performing the opposite operation: dividing both sides by 3. Think of it as unwinding a rope – you perform the reverse action to loosen the knots. In the same way, dividing both sides by 3 undoes the multiplication and allows us to isolate 'v'. This step is the culmination of all our previous efforts. It's the final piece of the puzzle that reveals the value of the variable we've been seeking. Now, let's see how dividing both sides by 3 works. On the left side, 3v / 3 simplifies to v. On the right side, 27 / 3 gives us 9. So, the equation 3v = 27 transforms into v = 9. Eureka! We've found the value of v. This transformation is the grand finale of our equation-solving journey. It's like reaching the summit of a mountain – you've overcome all the obstacles and achieved your goal. So, remember, whenever you need to solve for a variable, perform the opposite operation to isolate it. It's a powerful technique that unlocks the solution. With this step completed, we've successfully solved the equation. We've conquered the challenge and emerged victorious. So, let's celebrate our achievement and bask in the satisfaction of solving for v!

Dividing both sides by 3:

3v / 3 = 27 / 3

This gives us:

v = 9

Final Answer

So, the solution to the equation -2v + 5(v - 5) = 2 is v = 9. Great job, guys! You've successfully solved for v! Remember, the key to solving equations is to follow the steps carefully, maintain balance, and don't be afraid to ask for help when you need it. Keep practicing, and you'll become an equation-solving master in no time!

Therefore, v = 9

Keywords

  • Solve for v
  • Linear equation
  • Distributive property
  • Combine like terms
  • Isolate the variable
  • Algebra
  • Equation solving
  • Mathematical problem
  • Step-by-step guide
  • Equation balance

Repair Input Keyword

Solve the equation -2v + 5(v - 5) = 2 for v. Simplify the answer.