Solving For P: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into a classic algebra problem: solving for a variable, specifically 'p' in the equation $\frac{2}{3}-2 p=5+\frac{4}{3} p$. Don't worry, it's not as scary as it looks. We'll break it down into easy-to-follow steps, so you can confidently tackle these types of equations. Let's get started, shall we? This problem is a great example of a linear equation, and the goal is to isolate the variable 'p' on one side of the equation. This involves using inverse operations to undo the mathematical operations that are applied to 'p'. The process includes combining like terms, which means grouping terms that have the same variable and constant terms together. Understanding and mastering these concepts is fundamental to success in algebra and beyond. This is like building a solid foundation for more complex mathematical concepts you'll encounter later on. We will explore each step in detail to ensure you grasp every aspect. Ready to unlock the mystery of 'p'? Let's go!

Step 1: Combine the 'p' terms

Our first order of business when we solve for p in the equation $\frac{2}{3}-2 p=5+\frac{4}{3} p$ is to get all the terms containing 'p' on one side of the equation. To do this, we'll work with the equation and use our knowledge of inverse operations. Remember that the ultimate goal is to isolate 'p'. Now, we've got a '-2p' on the left side and a '+$\frac{4}{3} p$' on the right side. The standard approach is to move the 'p' term from the right side to the left side. To move the '+$\frac{4}{3} p$' from the right side, we perform the inverse operation: we subtract $\frac{4}{3} p$ from both sides of the equation. So, the equation $\frac{2}{3}-2 p=5+\frac{4}{3} p$ becomes:

$\frac{2}{3} - 2p - \frac{4}{3}p = 5 + \frac{4}{3}p - \frac{4}{3}p$

This simplifies to:

$\frac{2}{3} - 2p - \frac{4}{3}p = 5$

Now, let's combine the 'p' terms on the left side. Notice that both '-2p' and '-$\frac{4}{3}p$' are negative. This means we are adding the two negative terms. We can write -2 as -$\frac{6}{3}$ (because -2 is the same as -$\frac{6}{3}$). So, the equation becomes:

$\frac{2}{3} - \frac{6}{3}p - \frac{4}{3}p = 5$

Now, add the 'p' terms: -$\frac{6}{3}p - \frac{4}{3}p = -\frac{10}{3}p$. Our equation is now:

$\frac{2}{3} - \frac{10}{3}p = 5$

We've successfully moved the 'p' terms to one side of the equation. Keep up the good work; you're doing great!

Step 2: Isolate the 'p' term

Now that we've combined all the 'p' terms on one side in the equation $\frac{2}{3} - \frac{10}{3}p = 5$, the next step is to isolate the term containing 'p'. This means getting the '-$\frac{10}{3}p$' by itself on the left side of the equation. Currently, we have a $\frac{2}{3}$ added to '-$\frac{10}{3}p$'. To isolate the 'p' term, we need to eliminate this $\frac{2}{3}$. We do this by performing the inverse operation: subtracting $\frac{2}{3}$ from both sides of the equation. This ensures that the equation remains balanced.

So, the equation $\frac{2}{3} - \frac{10}{3}p = 5$ becomes:

$\frac{2}{3} - \frac{10}{3}p - \frac{2}{3} = 5 - \frac{2}{3}$

This simplifies to:

$- \frac{10}{3}p = 5 - \frac{2}{3}$

To subtract $\frac{2}{3}$ from 5, we first convert 5 into a fraction with a denominator of 3. We can write 5 as $\frac{15}{3}$ (because 5 is the same as $\frac{15}{3}$). Our equation becomes:

$- \frac{10}{3}p = \frac{15}{3} - \frac{2}{3}$

Subtract the fractions: $\frac{15}{3} - \frac{2}{3} = \frac{13}{3}$. Thus:

$- \frac{10}{3}p = \frac{13}{3}$

We're getting closer to solving for 'p'! Now the 'p' term is almost isolated.

Step 3: Solve for 'p' Final Step

We're in the final stretch now! We've made great progress in solving for 'p', and our equation currently looks like this: $- \frac{10}{3}p = \frac{13}{3}$. Our last step is to isolate 'p' completely. Right now, 'p' is being multiplied by -$\frac{10}{3}$. To isolate 'p', we need to undo this multiplication. We do this by dividing both sides of the equation by -$\frac{10}{3}$. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of -$\frac{10}{3}$ is -$\frac{3}{10}$. So, we'll multiply both sides of the equation by -$\frac{3}{10}$.

So, the equation $- \frac{10}{3}p = \frac{13}{3}$ becomes:

$- \frac{3}{10} * (-\frac{10}{3}p) = \frac{13}{3} * -\frac{3}{10}$

On the left side, -$\frac{3}{10}$ and -$\frac{10}{3}$ cancel out, leaving us with just 'p'.

$p = \frac{13}{3} * -\frac{3}{10}$

On the right side, multiply the numerators and the denominators: $\frac{13 * -3}{3 * 10} = \frac{-39}{30}$. This can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 3. So, $\frac{-39}{30} = -\frac{13}{10}$.

Therefore, we have:

$p = -\frac{13}{10}$

And there you have it! We've successfully solved for p in the equation $\frac{2}{3}-2 p=5+\frac{4}{3} p$, and the answer is $p = -\frac{13}{10}$. High five, you've done awesome. Keep practicing, and these problems will become second nature to you. Math can be fun when you break it down step by step!

Verification and Tips

Before we wrap things up, it's always a good idea to verify your answer to solve for p. We can plug our solution, $p = -\frac{13}{10}$, back into the original equation to ensure it holds true. This is a crucial step to avoid careless errors. Let's substitute $- \frac{13}{10}$ for 'p' in our original equation, $\frac{2}{3}-2 p=5+\frac{4}{3} p$.

$\frac{2}{3} - 2(-\frac{13}{10}) = 5 + \frac{4}{3}(-\frac{13}{10})$

Simplify the equation:

$\frac{2}{3} + \frac{26}{10} = 5 - \frac{52}{30}$

Further simplification gives us:

$\frac{2}{3} + \frac{13}{5} = 5 - \frac{26}{15}$

To add the fractions, find a common denominator (15 in this case):

$\frac{10}{15} + \frac{39}{15} = \frac{75}{15} - \frac{26}{15}$

$\frac{49}{15} = \frac{49}{15}$

Since both sides of the equation are equal, our solution $p = -\frac{13}{10}$ is correct. We can confidently say we've solved for 'p'. Some handy tips for solving similar problems: Always double-check your arithmetic, especially when dealing with fractions and negative numbers. Write each step clearly to minimize mistakes. Don't skip steps; each step builds on the previous one. Practice, practice, practice! The more you work through these problems, the more comfortable and confident you'll become. Remember to combine like terms carefully and perform inverse operations to isolate the variable. Finally, always verify your solution to ensure its accuracy. Keep up the excellent work, and happy solving!