Solving 3p = 24: A Simple Algebraic Equation

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Have you ever stumbled upon a seemingly simple equation like 3p = 24 and wondered how to solve it? Don't worry, you're not alone! Algebraic equations are a fundamental part of mathematics, and mastering them opens doors to more complex problem-solving. In this comprehensive guide, we'll break down the steps to solve 3p = 24, ensuring you grasp the underlying concepts and can confidently tackle similar problems in the future. So, let's dive in and unravel the mystery of this equation!

Understanding the Basics of Algebraic Equations

Before we jump into solving 3p = 24, it's crucial to understand the basics of algebraic equations. An algebraic equation is a mathematical statement that shows the equality between two expressions. These expressions often contain variables, which are symbols (usually letters) that represent unknown values. In our equation, 3p = 24, 'p' is the variable we need to find. The equation essentially asks: "What value of 'p' will make the left side (3p) equal to the right side (24)?"

Key components of an algebraic equation include:

  • Variables: Symbols (like 'p', 'x', or 'y') that represent unknown values.
  • Coefficients: Numbers that multiply variables (in 3p, '3' is the coefficient).
  • Constants: Numbers that stand alone (in 3p = 24, '24' is the constant).
  • Operations: Mathematical processes like addition, subtraction, multiplication, and division.

The goal of solving an equation is to isolate the variable on one side of the equation. This means manipulating the equation using mathematical operations until we have the variable by itself, revealing its value. Remember, whatever operation you perform on one side of the equation, you must also perform on the other side to maintain the equality. This principle is the cornerstone of solving algebraic equations.

Step-by-Step Solution to 3p = 24

Now, let's get down to business and solve our equation, 3p = 24. Here's a step-by-step approach:

Step 1: Identify the Operation

In the equation 3p = 24, '3p' means '3 multiplied by p'. So, the operation connecting the coefficient (3) and the variable (p) is multiplication.

Step 2: Apply the Inverse Operation

To isolate 'p', we need to undo the multiplication. The inverse operation of multiplication is division. Therefore, we'll divide both sides of the equation by the coefficient of 'p', which is 3.

Step 3: Perform the Division

Dividing both sides by 3, we get:

(3p) / 3 = 24 / 3

Step 4: Simplify the Equation

On the left side, 3 divided by 3 cancels out, leaving us with just 'p'. On the right side, 24 divided by 3 equals 8. So, our simplified equation is:

p = 8

Step 5: Verify the Solution (Optional but Recommended)

To ensure our solution is correct, we can substitute 'p = 8' back into the original equation and see if it holds true:

3 * 8 = 24

24 = 24

Since the equation is true, our solution is correct! We have successfully solved the equation 3p = 24, and the value of 'p' is 8.

Why This Works: The Principle of Inverse Operations

The core of solving algebraic equations lies in the principle of inverse operations. Each mathematical operation has an inverse that undoes it. Here's a quick recap:

  • The inverse of addition is subtraction.
  • The inverse of subtraction is addition.
  • The inverse of multiplication is division.
  • The inverse of division is multiplication.

By applying the inverse operation, we effectively "undo" the operation that's keeping the variable from being isolated. In our case, dividing by 3 was the inverse operation that allowed us to isolate 'p'.

Practice Makes Perfect: Examples and Exercises

Now that you understand the process, let's look at a few more examples and exercises to solidify your understanding.

Example 1: Solve 5x = 35

  1. Identify the operation: Multiplication (5 multiplied by x)
  2. Apply the inverse operation: Divide both sides by 5
  3. Perform the division: (5x) / 5 = 35 / 5
  4. Simplify: x = 7
  5. Verify: 5 * 7 = 35 (Correct!)

Example 2: Solve 2y + 4 = 10

This equation has two operations: multiplication (2y) and addition (+ 4). We need to undo them in the reverse order of operations (PEMDAS/BODMAS).

  1. Undo addition: Subtract 4 from both sides: 2y + 4 - 4 = 10 - 4
  2. Simplify: 2y = 6
  3. Undo multiplication: Divide both sides by 2: (2y) / 2 = 6 / 2
  4. Simplify: y = 3
  5. Verify: 2 * 3 + 4 = 10 (Correct!)

Exercises:

  1. Solve 4q = 28
  2. Solve 6z - 2 = 16
  3. Solve 10a = 100

(Answers at the end of this section)

Common Mistakes to Avoid

While solving algebraic equations is relatively straightforward, there are a few common mistakes to watch out for:

  • Forgetting to perform the operation on both sides: Remember, you must always maintain the equality by doing the same thing to both sides of the equation.
  • Incorrectly applying inverse operations: Make sure you're using the correct inverse operation (e.g., dividing instead of multiplying).
  • Not following the order of operations: When an equation has multiple operations, follow the reverse order of operations (PEMDAS/BODMAS) to solve it correctly.
  • Skipping the verification step: Always verify your solution by substituting it back into the original equation to catch any errors.

By being mindful of these common pitfalls, you can improve your accuracy and confidence in solving algebraic equations.

Real-World Applications of Algebraic Equations

Algebraic equations aren't just abstract mathematical concepts; they have numerous real-world applications. Here are a few examples:

  • Calculating distances, speeds, and times: If you know the speed of a car and the time it travels, you can use an equation to calculate the distance.
  • Determining costs and prices: Businesses use equations to calculate the cost of goods, set prices, and determine profits.
  • Converting units: Equations are used to convert between different units of measurement, such as feet to meters or Celsius to Fahrenheit.
  • Solving problems in science and engineering: Many scientific and engineering problems involve equations that need to be solved to find unknown quantities.

The ability to solve algebraic equations is a valuable skill that can be applied in various aspects of life, from everyday tasks to professional endeavors.

Conclusion: Mastering the Art of Solving Equations

Solving the equation 3p = 24 is just the tip of the iceberg when it comes to algebraic equations. By understanding the basic principles, practicing regularly, and avoiding common mistakes, you can master the art of solving equations and unlock a world of mathematical possibilities. Remember, mathematics is a journey of discovery, and each equation you solve is a step forward on that path. So, embrace the challenge, keep practicing, and watch your problem-solving skills soar!

Answers to Exercises:

  1. q = 7
  2. z = 3
  3. a = 10

For further learning and practice, you might find helpful resources at websites like Khan Academy Algebra.