Solving For 'v': A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the equation $\frac{4}{5}(v-7)=2$. Don't worry, it looks a little intimidating, but trust me, we'll break it down into easy-to-digest steps. Our mission is to find the value of 'v' that makes this equation true. Think of it like a treasure hunt; we're searching for the hidden 'v'. This is a fundamental concept in algebra, and understanding how to solve these types of equations is super important as you progress in math. This isn't just about getting an answer; it's about understanding the process of how to manipulate equations and isolate a variable. Ready to unlock the mystery of 'v'? Let's jump in! We will start with a review of basic algebraic principles to make sure we're all on the same page. Then, we'll take it step by step, explaining each action and why we're taking it. Finally, we'll check our answer to make certain we found the correct value for "v". Let's get started!

First, let's brush up on some key concepts. In algebra, we're constantly working with equations, which are mathematical statements that two expressions are equal. An equation contains an equals sign (=), and our goal is often to solve for a variable, like our 'v' here. This means finding the value that makes the equation true. To do this, we use various properties of equality. One of the most important is the property that says if we perform the same operation on both sides of an equation, the equation remains balanced. This is our golden rule! For instance, if we add 5 to both sides, subtract 10 from both sides, multiply by 2 on both sides, or divide by any non-zero number on both sides, the equality holds. The idea is to isolate the variable, get it all by itself on one side of the equation. This is achieved by systematically undoing any operations that have been performed on the variable. We do this in the reverse order of operations (PEMDAS/BODMAS): we deal with addition/subtraction first, then multiplication/division, and finally, exponents/parentheses if they're present. When we're given an equation like this, we'll often need to use the distributive property, which is another crucial tool. This property states that $a(b + c) = ab + ac$. It allows us to remove parentheses by multiplying the term outside the parentheses by each term inside. We'll encounter it in the next step. So, before proceeding, make sure you know the fundamentals. With these tools in our toolkit, we are well-prepared to tackle our equation and discover the value of 'v'.

Step-by-Step Solution

Alright, buckle up, because here comes the good part. Let's solve the equation $\frac{4}{5}(v-7)=2$ step by step.

Step 1: Eliminate the Fraction

Our initial equation is $\frac{4}{5}(v-7)=2$. The fraction $\frac{4}{5}$ might seem annoying, so let's get rid of it. We can do this by multiplying both sides of the equation by the reciprocal of $\frac{4}{5}$, which is $\frac{5}{4}$. Remember, what we do to one side, we must do to the other to keep things balanced!

So, we have: $\frac{5}{4} \cdot \frac{4}{5}(v-7) = 2 \cdot \frac{5}{4}$.

On the left side, $\frac{5}{4}$ and $\frac{4}{5}$ cancel each other out, leaving us with $(v-7)$. On the right side, $2 \cdot \frac{5}{4} = \frac{10}{4}$, which simplifies to $\frac{5}{2}$.

Now our equation is $(v-7) = \frac{5}{2}$.

Step 2: Isolate 'v'

Now that we've cleared the fraction, we want to isolate 'v'. We need to get rid of the -7 that's hanging out with our variable. To do this, we'll add 7 to both sides of the equation.

So, we get: $v-7 + 7 = \frac{5}{2} + 7$.

On the left side, $-7 + 7$ cancels out, leaving us with just 'v'. On the right side, we need to add $\frac{5}{2}$ and 7. To add these, we need a common denominator. Let's rewrite 7 as $\frac{14}{2}$.

Therefore, we have: $v = \frac{5}{2} + \frac{14}{2}$.

Step 3: Calculate the Final Value

Finally, let's add the two fractions on the right side: $v = \frac{5+14}{2}$.

This simplifies to: $v = \frac{19}{2}$.

And there you have it! We've found the value of 'v'.

Checking Your Answer

Always, and I mean always, double-check your answer! It's the best way to make sure you haven't made any sneaky mistakes. Let's plug our value for 'v' back into the original equation and see if it holds true.

Our original equation was $\frac{4}{5}(v-7)=2$, and we found that $v = \frac{19}{2}$.

Let's substitute $\frac{19}{2}$ for 'v': $\frac{4}{5}(\frac{19}{2}-7)=2$.

First, solve inside the parentheses: $\frac{19}{2} - 7$. Again, we need to rewrite 7 as $\frac{14}{2}$, so we have $\frac{19}{2} - \frac{14}{2} = \frac{5}{2}$.

Now the equation is: $\frac{4}{5}(\frac{5}{2})=2$.

Multiply the fractions: $\frac{4 \cdot 5}{5 \cdot 2} = \frac{20}{10} = 2$.

So, we have $2=2$. Our equation holds true! This means that our solution, $v = \frac{19}{2}$, is correct. Pat yourselves on the back, folks. That's a wrap! Solving equations like this is a fundamental skill, and you've just proven you've got it. Keep practicing, and you'll become a pro in no time.

Further Practice and Resources

Practice makes perfect, right? To really solidify your skills, try solving similar equations. You can change the numbers, introduce new variables, and experiment with different operations. The more you work with these types of problems, the more comfortable and confident you'll become. Here are a few examples to get you started:

• $\frac{2}{3}(x+4) = 6$
• $\frac{1}{2}(y-3) = 5$
• $\frac{3}{4}(z+2) = 9$

Feel free to pause and test yourself to see how well you have understood. If you want more problems, you can also search online for algebra practice worksheets and resources. Many websites offer free practice problems, video tutorials, and step-by-step explanations, so do not hesitate to make the most of it. Also, if you're struggling with specific concepts, don't be afraid to ask for help. Reach out to your teacher, classmates, or online math communities. Math is a journey, and there are many people ready and willing to support you.

To better understand the subject, you can also explore some related topics. Mastering these concepts will allow you to address more complex mathematical problems: understanding the order of operations (PEMDAS/BODMAS), using the distributive property, working with fractions, and solving linear equations. Each concept builds on the other, so make sure to review those concepts if you are struggling with a specific part of the process. If you want to take your mathematical prowess to the next level, you can practice more complex algebra problems. These advanced topics are built on these basics that you just mastered! As a quick tip, always remember to show your work and check your answers. This will help you catch any errors and solidify your understanding of the concepts. Keep practicing, keep learning, and don't give up. With a little effort and persistence, you'll become a math whiz in no time. Congratulations, and keep up the great work!