Solving 8(r-4)-7(r-1)=2(r+3) A Step-by-Step Guide

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Solving equations is a fundamental skill in mathematics, and mastering it opens doors to more complex problem-solving. In this comprehensive guide, we will delve into the step-by-step process of solving the equation 8(r-4) - 7(r-1) = 2(r+3). Our approach will not only provide the solution but also illuminate the underlying principles, ensuring a deep understanding of equation-solving techniques. By the end of this guide, you will be equipped with the knowledge and confidence to tackle similar equations with ease. Let's embark on this mathematical journey together!

1. Understanding the Equation

Before diving into the solution, it's crucial to understand the equation's structure. The equation 8(r-4) - 7(r-1) = 2(r+3) is a linear equation in one variable, 'r'. It involves parentheses, which indicate multiplication, and the variable 'r' appears on both sides of the equation. To solve for 'r', our goal is to isolate 'r' on one side of the equation. This involves simplifying the equation by removing parentheses, combining like terms, and performing operations to maintain the equality. This initial step sets the stage for a clear and organized solution process. By grasping the nature of the equation, we can approach it strategically and avoid common pitfalls. The equation represents a balance, and our task is to manipulate it while preserving that balance, ultimately revealing the value of 'r' that satisfies the equation.

2. Expanding the Parentheses

The first step in simplifying the equation is to eliminate the parentheses. We achieve this by applying the distributive property, which states that a(b + c) = ab + ac. Let's apply this property to both sides of the equation 8(r-4) - 7(r-1) = 2(r+3).

  • For the term 8(r-4), we distribute the 8 to both 'r' and '-4', resulting in 8r - 32.
  • For the term -7(r-1), we distribute the -7 to both 'r' and '-1', resulting in -7r + 7. Note that multiplying a negative number by a negative number yields a positive number.
  • For the term 2(r+3), we distribute the 2 to both 'r' and '3', resulting in 2r + 6.

Substituting these expanded terms back into the equation, we get: 8r - 32 - 7r + 7 = 2r + 6. This step is crucial as it transforms the equation into a more manageable form by removing the parentheses and exposing the individual terms. By carefully applying the distributive property, we ensure that the equation remains balanced and that no terms are missed or incorrectly calculated. This meticulous approach is essential for accurate equation solving.

3. Combining Like Terms

Now that we have expanded the parentheses, the next step is to combine like terms on each side of the equation. Like terms are terms that have the same variable raised to the same power. In our equation, 8r - 32 - 7r + 7 = 2r + 6, we have 'r' terms and constant terms on the left side.

  • Combining the 'r' terms, we have 8r - 7r, which simplifies to 1r or simply 'r'.
  • Combining the constant terms, we have -32 + 7, which simplifies to -25.

Therefore, the left side of the equation simplifies to r - 25. The right side of the equation, 2r + 6, already has its like terms combined. Our equation now looks like this: r - 25 = 2r + 6. This step is vital for simplifying the equation and making it easier to isolate the variable. By combining like terms, we reduce the number of terms in the equation, making it more streamlined and less prone to errors in subsequent steps. This process of simplification is a key strategy in equation solving.

4. Isolating the Variable

To isolate the variable 'r', we need to move all terms containing 'r' to one side of the equation and all constant terms to the other side. Our current equation is r - 25 = 2r + 6. A common strategy is to move the term with the smaller coefficient of 'r' to the side with the larger coefficient. In this case, we will move the 'r' term from the left side to the right side.

  • Subtract 'r' from both sides of the equation: (r - 25) - r = (2r + 6) - r. This simplifies to -25 = r + 6.

Now, we need to isolate 'r' by moving the constant term from the right side to the left side.

  • Subtract 6 from both sides of the equation: -25 - 6 = (r + 6) - 6. This simplifies to -31 = r.

Therefore, we have isolated 'r' and found that r = -31. This process of isolating the variable is the core of equation solving. By performing the same operations on both sides of the equation, we maintain the balance and gradually work towards isolating the variable, ultimately revealing its value. This strategic manipulation of the equation is a testament to the power of algebraic principles.

5. Verifying the Solution

After obtaining a solution, it's always a good practice to verify it by substituting the value back into the original equation. This ensures that our solution is correct and that no errors were made during the solving process. Our original equation is 8(r-4) - 7(r-1) = 2(r+3), and our solution is r = -31.

  • Substitute r = -31 into the left side of the equation: 8(-31-4) - 7(-31-1) = 8(-35) - 7(-32) = -280 + 224 = -56.
  • Substitute r = -31 into the right side of the equation: 2(-31+3) = 2(-28) = -56.

Since both sides of the equation equal -56 when r = -31, our solution is correct. This verification step is crucial for building confidence in our solution and identifying any potential errors. It reinforces the importance of accuracy and attention to detail in equation solving. By verifying our solution, we complete the equation-solving process and ensure that we have arrived at the correct answer.

6. Conclusion

In conclusion, we have successfully solved the equation 8(r-4) - 7(r-1) = 2(r+3) and found that r = -31. We achieved this by following a step-by-step process that included expanding parentheses, combining like terms, isolating the variable, and verifying the solution. Each step was carefully explained to provide a clear understanding of the underlying principles. Solving equations is a fundamental skill in mathematics, and mastering it requires practice and a systematic approach. By understanding the concepts and techniques presented in this guide, you can confidently tackle similar equations and advance your mathematical abilities. Remember to always verify your solutions to ensure accuracy and build confidence in your problem-solving skills. With dedication and practice, you can excel in equation solving and unlock the power of mathematics.