Solving -4.6p - 6.3p + 3.9 = -9.18 A Step-by-Step Solution

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Understanding how to solve linear equations is a fundamental skill in algebra. This article will guide you through the process of solving the equation -4.6p - 6.3p + 3.9 = -9.18, providing a clear, step-by-step explanation to help you master this essential mathematical concept. We'll break down each step, ensuring you understand not just the how, but also the why behind each operation. Our goal is to empower you with the knowledge and confidence to tackle similar problems with ease.

1. Combining Like Terms: Simplifying the Equation

The first crucial step in solving this equation involves combining like terms. In the given equation, -4.6p and -6.3p are like terms because they both contain the variable p. Combining these terms simplifies the equation and makes it easier to work with. Let's dive into the process:

  • Identifying Like Terms: Like terms are terms that have the same variable raised to the same power. In our equation, -4.6p and -6.3p fit this description. The term 3.9 is a constant term and cannot be combined with the terms containing p.
  • Adding the Coefficients: To combine like terms, we add their coefficients. The coefficient is the number that multiplies the variable. In this case, we add -4.6 and -6.3. When adding negative numbers, it's like moving further to the left on the number line. -4.6 + (-6.3) = -10.9.
  • Rewriting the Equation: After combining the like terms, the equation becomes -10.9p + 3.9 = -9.18. This simplified equation is much easier to solve than the original.

This step is crucial because it reduces the complexity of the equation, making subsequent steps more manageable. By combining like terms, we consolidate the variable terms into a single term, paving the way for isolating the variable and finding its value.

2. Isolating the Variable: Moving Towards the Solution

Now that we've simplified the equation to -10.9p + 3.9 = -9.18, the next step is to isolate the variable p. This means getting p by itself on one side of the equation. To do this, we need to eliminate the constant term (3.9) from the left side. The key principle here is to perform the same operation on both sides of the equation to maintain balance. Think of an equation as a scale – if you add or subtract something from one side, you must do the same to the other to keep it balanced.

  • Subtracting 3.9 from Both Sides: To eliminate 3.9 from the left side, we subtract it from both sides of the equation. This gives us: -10.9p + 3.9 - 3.9 = -9.18 - 3.9.
  • Simplifying the Equation: On the left side, 3.9 - 3.9 cancels out, leaving us with -10.9p. On the right side, we subtract 3.9 from -9.18. Subtracting a positive number from a negative number is like moving further to the left on the number line. -9.18 - 3.9 = -13.08.
  • The Resulting Equation: After this step, our equation is now -10.9p = -13.08. We're one step closer to isolating p.

This step highlights the importance of maintaining balance in an equation. By performing the same operation on both sides, we ensure that the equation remains true while moving closer to our solution. Subtracting 3.9 from both sides effectively isolates the term containing p, setting us up for the final step.

3. Solving for 'p': The Final Calculation

We've reached the final stage in solving for p. Our equation is currently -10.9p = -13.08. To isolate p completely, we need to get rid of the coefficient -10.9. Since -10.9 is multiplying p, we perform the inverse operation, which is division. Remember, whatever we do to one side of the equation, we must also do to the other to maintain balance.

  • Dividing Both Sides by -10.9: To isolate p, we divide both sides of the equation by -10.9. This gives us: -10.9p / -10.9 = -13.08 / -10.9.
  • Simplifying the Equation: On the left side, -10.9 / -10.9 cancels out, leaving us with just p. On the right side, we divide -13.08 by -10.9. A negative number divided by a negative number results in a positive number. -13.08 / -10.9 = 1.2.
  • The Solution: Therefore, the solution to the equation is p = 1.2.

This final step demonstrates the power of inverse operations in solving equations. By dividing both sides by the coefficient of p, we successfully isolated the variable and found its value. The solution p = 1.2 is the value that, when substituted back into the original equation, will make the equation true.

4. Verification: Ensuring the Solution is Correct

Before we declare victory, it's always a good practice to verify our solution. 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 us catch any potential errors and ensures that our solution is accurate. Let's plug p = 1.2 back into the original equation:

  • Original Equation: -4.6p - 6.3p + 3.9 = -9.18
  • Substituting p = 1.2: -4.6(1.2) - 6.3(1.2) + 3.9 = -9.18
  • Performing the Calculations: Now, we perform the multiplications and additions on the left side of the equation.
    • -4.6 * 1.2 = -5.52
    • -6.3 * 1.2 = -7.56
    • -5.52 - 7.56 + 3.9 = -9.18
  • Checking for Equality: The left side of the equation simplifies to -9.18, which is equal to the right side of the equation. This confirms that our solution p = 1.2 is correct.

Verification is a crucial step in problem-solving. It provides us with confidence in our answer and helps us develop a deeper understanding of the equation. By substituting the solution back into the original equation, we ensure that it satisfies the equation's conditions.

5. Conclusion: Mastering Linear Equations

In this article, we've walked through the process of solving the linear equation -4.6p - 6.3p + 3.9 = -9.18. We've covered essential steps such as combining like terms, isolating the variable, and verifying the solution. By understanding these steps, you can confidently tackle a wide range of linear equations. Remember, the key to mastering algebra is practice. The more you practice, the more comfortable you'll become with these concepts. Keep practicing, and you'll become a proficient problem solver!

  • Combine like terms: This simplifies the equation and makes it easier to work with.
  • Isolate the variable: This involves performing inverse operations to get the variable by itself on one side of the equation.
  • Solve for the variable: This involves performing the final calculations to find the value of the variable.
  • Verify the solution: This involves substituting the value back into the original equation to ensure it's correct.

By following these steps and practicing regularly, you can master the art of solving linear equations and excel in your algebra journey.

Therefore, the solution to the equation -4.6p - 6.3p + 3.9 = -9.18 is:

p = 1.2