Solving For 'p': A Linear Equation Explained

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Hey guys! Ever stumbled upon a linear equation and felt a bit lost on how to solve for a specific variable? Don't worry, you're not alone! Linear equations are a fundamental part of algebra, and mastering them opens doors to more complex mathematical concepts. In this guide, we'll break down the process of solving for a variable, using the equation 24p + 12 - 18p = 10 + 2p - 6 as our example. We'll walk through each step, making sure you understand the logic and techniques involved. So, grab your thinking caps, and let's dive in!

Understanding Linear Equations

Before we jump into solving for 'p', let's quickly recap what a linear equation actually is. At its core, a linear equation is an algebraic equation where the highest power of the variable is 1. Think of it as a straight line when graphed (hence the name "linear"). These equations typically involve variables (like our 'p'), constants (numbers), and mathematical operations such as addition, subtraction, multiplication, and division.

The goal when solving a linear equation is to isolate the variable you're interested in (in our case, 'p') on one side of the equation. This means manipulating the equation using valid algebraic operations until you have 'p' all by itself on one side, and a numerical value on the other side. That numerical value is the solution – the value of 'p' that makes the equation true.

Solving linear equations is a crucial skill in mathematics and has wide-ranging applications in various fields, from science and engineering to economics and finance. Being able to confidently solve these equations allows you to model real-world situations, make predictions, and solve problems effectively. So, understanding the underlying principles and techniques is definitely worth the effort!

Step-by-Step Solution for 24p + 12 - 18p = 10 + 2p - 6

Now, let's get our hands dirty and solve the equation 24p + 12 - 18p = 10 + 2p - 6 step by step. We'll break down each operation and explain the reasoning behind it.

1. Simplify Both Sides of the Equation

The first thing we want to do is simplify both sides of the equation as much as possible. This makes the equation easier to work with and reduces the chances of making errors. To simplify, we combine like terms. Like terms are terms that have the same variable raised to the same power (or are just constants).

On the left side of the equation, we have two terms with 'p': 24p and -18p. We can combine these by adding their coefficients: 24p - 18p = 6p. We also have a constant term, +12, which we'll keep as is for now. So, the left side simplifies to 6p + 12.

On the right side of the equation, we have a 'p' term (2p) and two constant terms: 10 and -6. Let's combine the constants: 10 - 6 = 4. So, the right side simplifies to 2p + 4.

After simplifying both sides, our equation now looks like this: 6p + 12 = 2p + 4.

2. Isolate the 'p' Terms on One Side

Our next goal is to get all the 'p' terms on one side of the equation. It doesn't matter which side we choose, but it's generally a good idea to move the term with the smaller coefficient to the side with the larger coefficient to avoid dealing with negative numbers (although it'll work either way!). In our case, 2p is smaller than 6p, so we'll move the 2p term to the left side.

To do this, we subtract 2p from both sides of the equation. Remember, whatever we do to one side of the equation, we must do to the other side to maintain the balance. Subtracting 2p from both sides gives us:

6p + 12 - 2p = 2p + 4 - 2p

Simplifying this, we get: 4p + 12 = 4

Now, all the 'p' terms are on the left side!

3. Isolate the Constant Terms on the Other Side

Next, we want to isolate the constant terms (the numbers without 'p') on the right side of the equation. We currently have a +12 on the left side, so we need to get rid of it. To do this, we subtract 12 from both sides of the equation:

4p + 12 - 12 = 4 - 12

Simplifying this, we get: 4p = -8

Now, we have all the 'p' terms on the left and all the constant terms on the right.

4. Solve for 'p'

Finally, we're ready to solve for 'p'. We have 4p = -8, which means 4 times 'p' is equal to -8. To isolate 'p', we need to undo the multiplication. We do this by dividing both sides of the equation by 4:

4p / 4 = -8 / 4

Simplifying this, we get: p = -2

And there you have it! We've solved for 'p'.

Verification: Plugging the Solution Back In

To make sure our solution is correct, it's always a good idea to plug the value we found for 'p' back into the original equation and see if it holds true. This is called verification.

Our original equation was: 24p + 12 - 18p = 10 + 2p - 6

We found that p = -2. Let's substitute -2 for 'p' in the equation:

24(-2) + 12 - 18(-2) = 10 + 2(-2) - 6

Now, we simplify both sides:

-48 + 12 + 36 = 10 - 4 - 6

0 = 0

Since both sides of the equation are equal when we substitute p = -2, our solution is correct! Awesome!

Key Takeaways and Tips for Solving Linear Equations

Solving linear equations might seem tricky at first, but with practice, it becomes second nature. Here are some key takeaways and tips to help you master the art of solving for variables:

  • Simplify, Simplify, Simplify: Always simplify both sides of the equation before doing anything else. Combine like terms to make the equation easier to manage.
  • Isolate the Variable: The main goal is to get the variable you're solving for all by itself on one side of the equation. Use inverse operations (addition/subtraction, multiplication/division) to move terms around.
  • Maintain Balance: Remember the golden rule: whatever you do to one side of the equation, you must do to the other side. This keeps the equation balanced and ensures you get the correct solution.
  • Verify Your Solution: Always plug your solution back into the original equation to check if it's correct. This is a great way to catch any mistakes.
  • Practice Makes Perfect: The more you practice solving linear equations, the better you'll become. Start with simple equations and gradually work your way up to more complex ones.

Common Mistakes to Avoid

While solving linear equations, it's easy to make small mistakes that can lead to incorrect answers. Here are some common pitfalls to watch out for:

  • Forgetting the Sign: Be careful with signs (positive and negative). A simple sign error can throw off your entire solution. Pay close attention when adding, subtracting, multiplying, or dividing negative numbers.
  • Incorrectly Combining Like Terms: Make sure you're only combining terms that are actually "like" (have the same variable raised to the same power). Don't try to combine terms like 3x and 3x², for example.
  • Not Distributing Properly: When you have a number multiplying a group of terms inside parentheses, remember to distribute the number to each term inside the parentheses. For example, 2(x + 3) becomes 2x + 6, not 2x + 3.
  • Dividing by Zero: Remember, you can never divide by zero! If you end up with an equation where you need to divide by zero, it means there's no solution (or infinitely many solutions, depending on the situation).
  • Not Checking Your Work: As we mentioned earlier, verifying your solution is crucial. It's a quick way to catch errors and build confidence in your answer.

By being aware of these common mistakes and taking the time to double-check your work, you can significantly reduce the chances of errors and become a more accurate problem-solver.

Practice Problems

Now that we've walked through the solution and covered some key tips, it's time to put your knowledge to the test! Here are a few practice problems for you to try. Remember to follow the steps we outlined above, and don't forget to verify your answers.

  1. 3x + 5 = 14
  2. 2(y - 1) = 8
  3. 5a - 7 = 3a + 1
  4. -4m + 9 = 21
  5. 7z + 2 = 9z - 4

The solutions to these problems are provided at the end of this guide, but try to solve them on your own first. The key is to practice!

Conclusion

Solving linear equations is a fundamental skill in algebra, and mastering it will set you up for success in more advanced mathematical topics. By understanding the principles of simplifying, isolating variables, and maintaining balance, you can confidently tackle a wide range of linear equations. Remember to practice regularly, verify your solutions, and be mindful of common mistakes. With consistent effort, you'll become a linear equation-solving pro!

So, the next time you encounter a linear equation, don't shy away. Embrace the challenge, put your skills to work, and solve for that variable! You've got this!

Solutions to Practice Problems:

  1. x = 3
  2. y = 5
  3. a = 4
  4. m = -3
  5. z = 3