Solving For P: 13p - 4(10 + 6p) = 4 | Step-by-Step Guide

by Andrew McMorgan 57 views

Hey guys! Let's dive into solving this algebraic equation together. If you're scratching your head over how to isolate 'p' in this equation, you've come to the right place. We'll break it down step-by-step, making it super easy to follow along. So, grab your pencils, and let's get started!

Understanding the Equation

The equation we're tackling today is: 13p - 4(10 + 6p) = 4. This looks a bit intimidating at first, but don’t worry! We'll use basic algebraic principles to simplify and solve it. The key is to remember the order of operations (PEMDAS/BODMAS) and apply the distributive property correctly. We want to isolate the variable 'p' on one side of the equation to find its value. So, let’s jump into the step-by-step solution and make this equation our algebraic playground.

Step 1: Distribute the -4

The first thing we need to do is get rid of those parentheses. Remember the distributive property? It's where we multiply the term outside the parentheses by each term inside. In our case, we're distributing -4 across (10 + 6p).

So, -4 multiplied by 10 is -40, and -4 multiplied by 6p is -24p. Our equation now looks like this:

13p - 40 - 24p = 4

This step is crucial because it simplifies the equation and allows us to combine like terms. Remember to pay close attention to the signs (positive or negative) when you distribute! This part is super important for getting the correct answer, so double-check your work here. Now that we've distributed, the equation is much easier to handle. We're one step closer to solving for 'p'.

Step 2: Combine Like Terms

Now that we've distributed, let's tidy things up by combining the like terms on the left side of the equation. Look for terms that have the same variable ('p' in this case) or are constants (just numbers). We have two terms with 'p': 13p and -24p. Combining these gives us:

13p - 24p = -11p

So, our equation now looks like this:

-11p - 40 = 4

By combining like terms, we've simplified the equation further, making it even more manageable. This step is like organizing your workspace before tackling a big project—it helps clear the clutter and makes the next steps much clearer. We're making great progress in isolating 'p'! Next up, we'll move the constant term to the other side of the equation.

Step 3: Isolate the Variable Term

Our next goal is to get the term with 'p' all by itself on one side of the equation. Right now, we have “-11p - 40 = 4”. We need to get rid of that “- 40”. How do we do that? By performing the opposite operation! Since we're subtracting 40, we'll add 40 to both sides of the equation. This keeps the equation balanced, which is super important in algebra.

So, let's add 40 to both sides:

-11p - 40 + 40 = 4 + 40

This simplifies to:

-11p = 44

Awesome! We've successfully isolated the variable term (-11p) on the left side. We're getting closer and closer to finding the value of 'p'. Just one more step to go!

Step 4: Solve for p

We're almost there! We now have -11p = 44. To finally solve for 'p', we need to get 'p' all by itself. Right now, 'p' is being multiplied by -11. So, to undo that multiplication, we'll do the opposite operation: division. We'll divide both sides of the equation by -11.

Let's do it:

-11p / -11 = 44 / -11

This simplifies to:

p = -4

And there you have it! We've solved for 'p'. The value of 'p' that makes the equation true is -4. Give yourself a pat on the back—you've just conquered an algebraic equation!

Verifying the Solution

To make sure we've got the correct answer, it's always a good idea to plug our solution back into the original equation. This is like double-checking your work to ensure everything adds up. Our original equation was:

13p - 4(10 + 6p) = 4

Let's substitute p = -4 into the equation:

13(-4) - 4(10 + 6(-4)) = 4

Now, we'll simplify:

-52 - 4(10 - 24) = 4

-52 - 4(-14) = 4

-52 + 56 = 4

4 = 4

It checks out! Both sides of the equation are equal, which means our solution p = -4 is correct. Verifying your solution is a great way to build confidence in your answer and make sure you haven't made any mistakes along the way.

Key Takeaways

Let's recap the key steps we took to solve this equation. This will help reinforce the process and make it easier to tackle similar problems in the future.

  1. Distribute: We started by distributing the -4 across the terms inside the parentheses. This cleared the parentheses and made the equation easier to work with.
  2. Combine Like Terms: Next, we combined like terms on the left side of the equation. This simplified the equation further and reduced the number of terms.
  3. Isolate the Variable Term: We then isolated the term with 'p' by adding 40 to both sides of the equation. This moved the constant term to the other side and brought us closer to solving for 'p'.
  4. Solve for p: Finally, we solved for 'p' by dividing both sides of the equation by -11. This gave us the value of 'p' that makes the equation true.
  5. Verify the Solution: We plugged our solution back into the original equation to double-check our work and ensure we got the correct answer.

By following these steps, you can confidently solve algebraic equations like this one. Remember, practice makes perfect, so keep at it!

Tips for Solving Equations

Here are some extra tips to keep in mind when solving algebraic equations. These tips can help you avoid common mistakes and improve your problem-solving skills.

  • Always follow the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). This ensures you simplify expressions in the correct order.
  • Pay attention to signs: Be careful with positive and negative signs. A small mistake with a sign can lead to a wrong answer. Double-check your signs throughout the process.
  • Perform the same operation on both sides: To keep the equation balanced, always perform the same operation on both sides. Whether it's adding, subtracting, multiplying, or dividing, make sure you do it to both sides.
  • Double-check your work: After you've found a solution, plug it back into the original equation to verify that it's correct. This is a great way to catch any mistakes.
  • Practice regularly: The more you practice, the better you'll become at solving equations. Set aside some time each day or week to work on algebra problems.

Common Mistakes to Avoid

Everyone makes mistakes sometimes, but knowing common pitfalls can help you avoid them. Here are a few mistakes to watch out for when solving equations:

  • Incorrectly distributing: Make sure you multiply the term outside the parentheses by every term inside the parentheses. Don't forget to pay attention to the signs!
  • Combining unlike terms: You can only combine terms that have the same variable and exponent (e.g., 3x and 5x can be combined, but 3x and 5x² cannot).
  • Forgetting to perform the same operation on both sides: If you add, subtract, multiply, or divide on one side of the equation, you must do the same on the other side to keep it balanced.
  • Making sign errors: Be extra careful with positive and negative signs. A small mistake can throw off your entire solution.
  • Not verifying the solution: Always plug your solution back into the original equation to make sure it's correct. This can help you catch any mistakes you might have made.

Practice Problems

Now that we've gone through the steps and tips, let's try some practice problems to reinforce what we've learned. Working through these problems will help you build confidence and improve your skills.

  1. Solve for x: 2x + 3(x - 4) = 6
  2. Solve for y: 5y - 2(y + 1) = 10
  3. Solve for a: 7a + 4(2a - 3) = 15

Try solving these problems on your own, using the steps we discussed earlier. Remember to distribute, combine like terms, isolate the variable term, solve for the variable, and verify your solution. The more you practice, the easier it will become!

Conclusion

So, we've successfully solved the equation 13p - 4(10 + 6p) = 4 and found that p = -4. We walked through each step, from distributing and combining like terms to isolating the variable and verifying our solution. You've learned some valuable skills for tackling algebraic equations!

Remember, solving equations is a fundamental skill in algebra, and the more you practice, the better you'll become. Keep these tips and strategies in mind as you continue your math journey. You've got this! Keep practicing and exploring new mathematical challenges. Until next time, happy solving!