Solving For P In The Equation 1/3 P + 1/2 P + 7 = 7/6 P + 5 + 4
Finding the solution for a variable in an equation is a fundamental skill in algebra. In this comprehensive guide, we'll break down the process of solving for p in the given equation:
1/3 p + 1/2 p + 7 = 7/6 p + 5 + 4
We will walk through each step meticulously, ensuring you grasp the underlying concepts and can confidently tackle similar problems. We'll start by simplifying the equation, combining like terms, isolating the variable p, and ultimately arriving at the correct solution. This guide aims to not only provide the answer but also to enhance your understanding of algebraic problem-solving techniques.
1. Simplifying the Equation: Combining Like Terms
To effectively solve for p, the first crucial step involves simplifying both sides of the equation. This means combining like terms, which are terms that have the same variable raised to the same power (in this case, just p) or are constants (numbers without variables). By simplifying, we make the equation more manageable and easier to work with. Let's delve into the process:
1.1 Combining Terms with p on the Left Side
On the left side of the equation, we have two terms containing p: 1/3 p and 1/2 p. To combine these, we need to find a common denominator for the fractions 1/3 and 1/2. The least common denominator (LCD) for 3 and 2 is 6. So, we convert both fractions to have a denominator of 6:
- 1/3 p = (1/3) * (2/2) p = 2/6 p
- 1/2 p = (1/2) * (3/3) p = 3/6 p
Now we can add these terms together:
2/6 p + 3/6 p = (2/6 + 3/6) p = 5/6 p
So, the left side of the equation now becomes:
5/6 p + 7
1.2 Combining Constant Terms on the Right Side
On the right side of the equation, we have two constant terms: 5 and 4. These are straightforward to combine:
5 + 4 = 9
So, the right side of the equation now becomes:
7/6 p + 9
1.3 The Simplified Equation
After combining like terms on both sides, our equation now looks like this:
5/6 p + 7 = 7/6 p + 9
This simplified equation is much easier to work with. We have reduced the number of terms and made the equation more concise. The next step is to isolate the terms with p on one side of the equation and the constant terms on the other. This will bring us closer to solving for p.
2. Isolating the Variable: Moving Terms Around
Now that we've simplified the equation, the next crucial step is to isolate the variable p. This means getting all the terms containing p on one side of the equation and all the constant terms (numbers) on the other side. This is achieved by performing the same operations on both sides of the equation, maintaining the balance and equality.
2.1 Moving p Terms to One Side
Currently, we have p terms on both sides of the equation:
5/6 p + 7 = 7/6 p + 9
To consolidate the p terms, we can subtract 5/6 p from both sides. This will eliminate the p term on the left side:
5/6 p + 7 - 5/6 p = 7/6 p + 9 - 5/6 p
This simplifies to:
7 = 2/6 p + 9
Notice that we subtracted 5/6 p from both 5/6 p and 7/6 p. On the left side, 5/6 p - 5/6 p cancels out, leaving just 7. On the right side, 7/6 p - 5/6 p equals 2/6 p (which can be further simplified to 1/3 p, but we'll keep it as 2/6 p for now).
2.2 Moving Constant Terms to the Other Side
Now we need to move the constant terms to the other side of the equation. We have the constant 9 on the right side, so we'll subtract 9 from both sides to eliminate it:
7 - 9 = 2/6 p + 9 - 9
This simplifies to:
-2 = 2/6 p
Now we have all the p terms on one side and the constant terms on the other. The equation is now in a form where we can easily solve for p. We've successfully isolated the variable, which is a significant step towards finding the solution.
3. Solving for p: Isolating the Variable Completely
After isolating the variable terms and constant terms on opposite sides of the equation, we've arrived at a point where we can directly solve for p. Our equation currently looks like this:
-2 = 2/6 p
To completely isolate p, we need to get rid of the coefficient (the number multiplying p), which in this case is 2/6. The most common way to do this is by multiplying both sides of the equation by the reciprocal of the coefficient. The reciprocal of 2/6 is 6/2, which simplifies to 3. Let's perform this operation:
3.1 Multiplying by the Reciprocal
Multiply both sides of the equation by 6/2 (or 3):
-2 * (6/2) = (2/6 p) * (6/2)
On the left side:
-2 * (6/2) = -2 * 3 = -6
On the right side, the fractions cancel out:
(2/6 p) * (6/2) = p
This leaves us with:
-6 = p
3.2 The Solution
Therefore, the solution for p in the equation is:
p = -6
We have successfully isolated p and found its value. This means that if we substitute -6 for p in the original equation, both sides of the equation will be equal. We've solved for p!
4. Verifying the Solution: Plugging It Back In
To ensure that our solution is correct, it's always a good practice to verify it. This involves substituting the value we found for p back into the original equation and checking if both sides of the equation are equal. This step helps catch any potential errors made during the solving process.
4.1 Substituting p = -6 into the Original Equation
Our original equation was:
1/3 p + 1/2 p + 7 = 7/6 p + 5 + 4
Substitute p = -6 into the equation:
1/3 (-6) + 1/2 (-6) + 7 = 7/6 (-6) + 5 + 4
Now, let's simplify each term:
- 1/3 (-6) = -2
- 1/2 (-6) = -3
- 7/6 (-6) = -7
So the equation becomes:
-2 + (-3) + 7 = -7 + 5 + 4
4.2 Simplifying Both Sides
Now, let's simplify both sides of the equation:
- Left side: -2 - 3 + 7 = 2
- Right side: -7 + 5 + 4 = 2
We see that both sides of the equation simplify to 2:
2 = 2
4.3 Conclusion: Solution Verified
Since both sides of the equation are equal when we substitute p = -6, our solution is correct. This verification step confirms that we have accurately solved for p and that our answer is valid. Verifying the solution is an essential part of the problem-solving process, especially in algebra, as it ensures the correctness of the final answer.
Conclusion: The Value of p
In this detailed guide, we have successfully solved for p in the equation:
1/3 p + 1/2 p + 7 = 7/6 p + 5 + 4
By systematically simplifying the equation, combining like terms, isolating the variable, and verifying our solution, we have determined that:
p = -6
This step-by-step approach not only provides the answer but also reinforces the fundamental principles of algebraic problem-solving. Understanding how to manipulate equations and isolate variables is crucial for success in mathematics and related fields. Remember, practice is key to mastering these skills. By working through similar problems, you can build confidence and proficiency in solving algebraic equations. The ability to solve for variables like p is a powerful tool that will serve you well in various mathematical contexts.
