Equation Solving: P - 3 1/6 = -2 1/2

Hey math enthusiasts! Today, we're diving into the world of algebraic equations to solve for our unknown, 'p'. Specifically, we're tackling the equation: p - 3 1/6 = -2 1/2. Solving equations like this is a fundamental skill in mathematics, opening doors to more complex problem-solving in various fields. Whether you're a student grappling with homework or just someone who enjoys a good mental workout, understanding how to isolate a variable is key. We'll break this down step-by-step, making sure that by the end, you'll feel confident in your ability to handle similar problems. Remember, practice makes perfect, and each equation you solve builds your mathematical muscles!

Understanding the Equation

Let's first get a clear picture of the equation we're working with: p - 3 1/6 = -2 1/2. Our goal is to find the value of 'p' that makes this statement true. In this equation, we have 'p' minus a mixed number (3 and one-sixth) equaling another mixed number, which is negative (negative 2 and one-half). Mixed numbers can sometimes feel a bit tricky, so a common first step in solving equations like this is to convert them into improper fractions. This often simplifies the arithmetic involved. An improper fraction has a numerator that is greater than or equal to its denominator, making it easier to work with when performing addition, subtraction, multiplication, or division. We'll convert both 3 1/6 and -2 1/2 into improper fractions. To convert a mixed number like a rac{b}{c} into an improper fraction, you multiply the whole number 'a' by the denominator 'c', and then add the numerator 'b'. The result becomes the new numerator, and the denominator 'c' stays the same. So, for 3 1/6, we multiply 3 by 6 to get 18, and then add 1 to get 19. The improper fraction is thus 19/6. For -2 1/2, we first convert 2 1/2. We multiply 2 by 2 to get 4, and add 1 to get 5. So, 2 1/2 becomes 5/2. Since we started with -2 1/2, our improper fraction will be -5/2. By performing this conversion, our equation now looks like: p - 19/6 = -5/2. This form often makes it much clearer how to proceed with isolating 'p'. We are essentially trying to undo the subtraction of 19/6 from 'p'.

Isolating the Variable 'p'

Now that we've rewritten our equation with improper fractions, p - 19/6 = -5/2, the next logical step is to isolate 'p' on one side of the equation. To do this, we need to get rid of the '- 19/6' term. The inverse operation of subtraction is addition. Therefore, to cancel out the '- 19/6', we must add 19/6 to both sides of the equation. This is a fundamental principle in algebra: whatever operation you perform on one side of the equation, you must perform the exact same operation on the other side to maintain the balance and equality. So, we'll add 19/6 to the left side and 19/6 to the right side: p - 19/6 + 19/6 = -5/2 + 19/6. On the left side, '- 19/6 + 19/6' cancels out to zero, leaving us with just 'p'. On the right side, we have a fraction addition problem: -5/2 + 19/6. To add fractions, they must have a common denominator. The denominators we have are 2 and 6. The least common multiple of 2 and 6 is 6. So, we need to convert -5/2 into an equivalent fraction with a denominator of 6. We can do this by multiplying both the numerator and the denominator by 3 (since 2 * 3 = 6). This gives us (-5 * 3) / (2 * 3) = -15/6. Now, our addition becomes: -15/6 + 19/6. With a common denominator, we can now add the numerators: -15 + 19 = 4. The denominator remains 6. So, the result of the right side is 4/6. Therefore, we have found that p = 4/6. This is our solution, but we can simplify it further.

Simplifying the Solution

We have arrived at the solution p = 4/6. While this is a correct value for 'p', it's always good practice in mathematics to simplify fractions to their lowest terms. A fraction is in its lowest terms when the greatest common divisor (GCD) of its numerator and denominator is 1. In the case of 4/6, both 4 and 6 are even numbers, which means they are both divisible by 2. The GCD of 4 and 6 is 2. To simplify the fraction, we divide both the numerator and the denominator by their GCD, which is 2. So, 4 divided by 2 is 2, and 6 divided by 2 is 3. This gives us the simplified fraction 2/3. Therefore, the final, simplified solution to our equation is p = 2/3. You can think of this as the simplest way to express the value of 'p'. To check our work, we can substitute p = 2/3 back into the original equation: (2/3) - 3 rac{1}{6} = -2 rac{1}{2}. Let's convert everything to improper fractions with a common denominator. We know 3 rac{1}{6} = rac{19}{6} and -2 rac{1}{2} = -rac{5}{2}. We also need to convert 2/3 to have a denominator of 6. Multiplying the numerator and denominator by 2, we get $(2 * 2) / (3 * 2) = 4/6$. So the equation becomes: $4/6 - 19/6 = -5/2$. On the left side, $4/6 - 19/6 = (4 - 19)/6 = -15/6$. Now we need to check if $-15/6$ is equal to $-5/2$. We can simplify $-15/6$ by dividing both numerator and denominator by their GCD, which is 3. $-15 ext{ divided by } 3 = -5$, and $6 ext{ divided by } 3 = 2$. So, $-15/6$ simplifies to $-5/2$. This matches the right side of our equation! This verification confirms that our solution p = 2/3 is indeed correct. This process of solving and checking is essential for building confidence in your mathematical abilities. If you're looking to further hone your equation-solving skills, resources like Khan Academy offer fantastic tutorials and practice problems on algebra and fractions.

Conclusion

In summary, we successfully solved the equation p - 3 1/6 = -2 1/2 by first converting the mixed numbers into improper fractions, resulting in p - 19/6 = -5/2. We then isolated 'p' by adding 19/6 to both sides of the equation, which required finding a common denominator for the fractions on the right side. After performing the fraction addition, we found that p = 4/6, which we then simplified to its lowest terms, p = 2/3. This step-by-step approach, involving conversion, inverse operations, and fraction arithmetic, is a robust method for solving linear equations. Remember that understanding fractions and their operations is a cornerstone of algebra. Consistent practice with these types of problems will significantly improve your mathematical fluency and problem-solving capabilities. Keep practicing, and don't be afraid to tackle more challenging equations as you grow more comfortable! For additional learning and practice on algebraic equations and fraction manipulation, you can explore resources like Paul's Online Math Notes.