Solve For P: A Step-by-Step Guide

by Sebastian Müller 34 views

Hey there, math enthusiasts! Today, we're diving into a fun little equation to solve for the variable p. This type of problem is a staple in algebra, and mastering it will definitely boost your math skills. We'll break it down step-by-step, so you can follow along easily. Let's get started!

Understanding the Equation

First, let's take a good look at the equation we're dealing with:

12 = (18/7)p

Solving for p in this equation means we want to isolate p on one side of the equals sign. In other words, we want to get p all by itself so we can see what value it represents. To do this, we need to undo the operations that are being performed on p. In this case, p is being multiplied by the fraction 18/7. So, what's the opposite of multiplication? Division, of course! But instead of dividing by a fraction, which can get a little messy, we're going to use a clever trick: multiplying by the reciprocal. The reciprocal of a fraction is simply flipping it upside down. So, the reciprocal of 18/7 is 7/18. Remember, the key to solving equations is to keep both sides balanced. Whatever we do to one side, we must do to the other. This ensures that the equality remains true. This principle is fundamental in algebra and is used extensively in solving various types of equations, from simple linear equations to more complex polynomial and trigonometric equations. Understanding and applying this principle correctly is crucial for success in algebra and beyond.

Step-by-Step Solution

Step 1: Multiply both sides by the reciprocal

Okay, let's multiply both sides of the equation by 7/18. This is the crucial step that will help us isolate p.

(7/18) * 12 = (7/18) * (18/7)p

On the right side, the (7/18) and (18/7) cancel each other out, leaving us with just p. This is exactly what we wanted! On the left side, we have (7/18) * 12. To multiply a fraction by a whole number, we can think of the whole number as a fraction with a denominator of 1. So, 12 is the same as 12/1. Now we multiply the numerators (7 * 12) and the denominators (18 * 1).

Step 2: Simplify the equation

Let's simplify both sides of the equation. On the left side, we have:

(7 * 12) / (18 * 1) = 84 / 18

We can simplify this fraction by finding the greatest common divisor (GCD) of 84 and 18. The GCD is the largest number that divides both 84 and 18 without leaving a remainder. In this case, the GCD is 6. So, we can divide both the numerator and the denominator by 6:

84 / 6 = 14
18 / 6 = 3

So, 84/18 simplifies to 14/3. On the right side of the equation, as we discussed earlier, the (7/18) and (18/7) cancel each other out, leaving us with just p. So our equation now looks like this:

14/3 = p

Step 3: State the solution

We've done it! We've successfully isolated p. Our solution is:

p = 14/3

This means that the value of p that makes the original equation true is 14/3. We can leave the answer as an improper fraction (14/3) or convert it to a mixed number. To convert 14/3 to a mixed number, we divide 14 by 3. 3 goes into 14 four times (4 * 3 = 12), with a remainder of 2. So, 14/3 is equal to 4 and 2/3. Either form of the answer is correct.

Verification

To be absolutely sure we got the correct answer, it's always a good idea to plug our solution back into the original equation and see if it holds true. This process is called verification, and it's a great way to catch any mistakes you might have made along the way. Let's substitute p = 14/3 back into the original equation:

12 = (18/7) * (14/3)

To multiply the fractions on the right side, we multiply the numerators (18 * 14) and the denominators (7 * 3):

(18 * 14) / (7 * 3) = 252 / 21

Now we simplify the fraction 252/21. We can divide both the numerator and the denominator by their greatest common divisor, which is 21:

252 / 21 = 12
21 / 21 = 1

So, 252/21 simplifies to 12/1, which is just 12. Now our equation looks like this:

12 = 12

This is a true statement! This confirms that our solution, p = 14/3, is indeed correct. Verification is a powerful tool in mathematics. It not only helps you check your answers but also deepens your understanding of the problem-solving process. By verifying your solutions, you build confidence in your work and develop a stronger intuition for mathematical concepts.

Alternative Method: Cross-Multiplication

There's another method we can use to solve this equation, called cross-multiplication. Cross-multiplication is a handy shortcut for solving equations where you have a fraction equal to a fraction. Our equation isn't quite in that form yet, but we can easily rewrite it to make it so. Remember that any whole number can be written as a fraction with a denominator of 1. So, we can rewrite 12 as 12/1. Now our equation looks like this:

12/1 = (18/7)p

To apply cross-multiplication, we multiply the numerator of the first fraction by the denominator of the second fraction, and set it equal to the product of the denominator of the first fraction and the numerator of the second fraction. In other words, we multiply diagonally.

Step 1: Rewrite the equation for cross-multiplication

First, let's rewrite the right side of the equation to make it look like a single fraction:

12/1 = (18p) / 7

Now we can clearly see the two fractions that we'll be cross-multiplying.

Step 2: Apply cross-multiplication

Cross-multiplying gives us:

12 * 7 = 1 * 18p

This simplifies to:

84 = 18p

Step 3: Isolate p

Now we need to isolate p. To do this, we divide both sides of the equation by 18:

84 / 18 = (18p) / 18

This simplifies to:

84/18 = p

Step 4: Simplify the fraction

We simplify the fraction 84/18 by dividing both the numerator and the denominator by their greatest common divisor, which is 6:

84 / 6 = 14
18 / 6 = 3

So, 84/18 simplifies to 14/3. Our solution is:

p = 14/3

This is the same solution we found using the first method, which gives us confidence that we're on the right track. Cross-multiplication is a valuable tool in your math arsenal. It's particularly useful when dealing with proportions and equations involving fractions. However, it's important to understand the underlying principles of equation solving so you can apply the correct methods in various situations.

Key Takeaways

  • To solve for a variable, we need to isolate it on one side of the equation.
  • We can use inverse operations to undo operations being performed on the variable.
  • Multiplying by the reciprocal is a useful technique for dividing by a fraction.
  • Always keep both sides of the equation balanced.
  • Verification is a great way to check your answers.
  • Cross-multiplication is a shortcut for solving equations with fractions.

Practice Makes Perfect

Solving equations is a fundamental skill in algebra, and the more you practice, the better you'll become. Try solving similar equations with different numbers and fractions. You can even create your own equations to challenge yourself! Remember, math is like a muscle – the more you use it, the stronger it gets. Keep practicing, and you'll become a pro at solving for variables in no time!

Keep up the awesome work, mathletes! You've got this!