Solving For P: 5(p-8)-(3p-12)=6 Explained!

by Alex Johnson 43 views

Let's break down how to solve the equation 5(p-8) - (3p-12) = 6 for the variable p. This is a common type of algebra problem that involves distribution, combining like terms, and isolating the variable. By following a step-by-step approach, we can find the value of p that satisfies the equation. Our goal is to manipulate the equation until we have p alone on one side, giving us the solution. So, grab your pencil and paper, and let's dive in!

Understanding the Equation

The equation we need to solve is 5(p-8) - (3p-12) = 6. This looks a little intimidating at first glance, but don't worry! We'll take it one step at a time. The left side of the equation involves expressions with the variable p, while the right side is a constant value, 6. The key to solving this equation is to simplify both sides by using the distributive property and combining like terms.

Breaking Down the Terms

First, we need to understand the different parts of the equation:

  • 5(p-8): This term means we need to multiply 5 by both p and -8.
  • (3p-12): This is a binomial expression containing a term with p and a constant.
  • 6: This is a constant term on the right side of the equation.

Our job is to simplify the equation by performing these operations correctly. Remember, the order of operations (PEMDAS/BODMAS) is crucial: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

Step-by-Step Solution

Now, let's walk through the solution step-by-step:

1. Distribute the 5 and the Negative Sign

The first thing we need to do is distribute the 5 in the term 5(p-8) and distribute the negative sign in front of (3p-12).

  • 5 * p = 5p
  • 5 * -8 = -40
  • -1 * 3p = -3p
  • -1 * -12 = +12

So, the equation becomes:

5p - 40 - 3p + 12 = 6

Important Note: Be very careful with the negative sign in front of the parentheses. It's like multiplying the entire expression inside the parentheses by -1.

2. Combine Like Terms

Next, we combine the like terms on the left side of the equation. Like terms are terms that have the same variable raised to the same power. In this case, we have terms with p (5p and -3p) and constant terms (-40 and +12).

  • Combine the p terms: 5p - 3p = 2p
  • Combine the constant terms: -40 + 12 = -28

So, the equation now looks like this:

2p - 28 = 6

3. Isolate the Variable Term

Our next goal is to isolate the term with p (2p) on one side of the equation. To do this, we need to get rid of the constant term (-28) on the left side. We can do this by adding 28 to both sides of the equation.

2p - 28 + 28 = 6 + 28

This simplifies to:

2p = 34

4. Solve for p

Finally, to solve for p, we need to get p by itself. Since p is being multiplied by 2, we can undo this multiplication by dividing both sides of the equation by 2.

2p / 2 = 34 / 2

This gives us:

p = 17

Verification

To make sure our solution is correct, we can substitute p = 17 back into the original equation and see if it holds true.

5(17 - 8) - (3 * 17 - 12) = 6

Let's simplify:

  • 5(9) - (51 - 12) = 6
  • 45 - (39) = 6
  • 45 - 39 = 6
  • 6 = 6

Since the equation holds true, our solution p = 17 is correct!

Common Mistakes to Avoid

  • Sign Errors: Be extra careful with negative signs, especially when distributing. A common mistake is forgetting to distribute the negative sign to all terms inside the parentheses.
  • Order of Operations: Always follow the order of operations (PEMDAS/BODMAS). Make sure to distribute before combining like terms.
  • Combining Incorrect Terms: Only combine like terms. For example, you can't combine a term with p with a constant term.
  • Forgetting to Distribute: When you have a number or a negative sign outside parentheses, make sure to distribute it to every term inside the parentheses.

Practice Problems

To improve your skills, try solving these similar equations:

  1. 3(x + 4) - (2x - 5) = 10
  2. 7(y - 2) + 4(2y + 1) = 15
  3. 2(a + 6) - 5(a - 3) = 9

Solving these types of equations involves similar steps: distribution, combining like terms, and isolating the variable. Practice makes perfect!

Tips for Success

  • Write Clearly: Write each step clearly and neatly. This will help you avoid mistakes and make it easier to follow your work.
  • Check Your Work: After each step, double-check your work to make sure you haven't made any errors.
  • Show All Steps: Even if you can do some steps in your head, it's a good idea to write them out. This will help you catch mistakes and make it easier for others to understand your solution.
  • Stay Organized: Keep your work organized and easy to read. This will help you stay focused and avoid confusion.

Conclusion

Solving equations like 5(p-8) - (3p-12) = 6 involves a few key steps: distributing, combining like terms, isolating the variable, and verifying your solution. By mastering these steps, you'll be well on your way to solving more complex algebraic equations. Remember to be careful with negative signs, follow the order of operations, and always check your work. Keep practicing, and you'll become more confident in your ability to solve for p or any other variable!

For further learning, check out Khan Academy's Algebra Resources for more in-depth explanations and practice problems.