Solving For N The Equation (3n + 3)/5 = (5n - 1)/9

by ADMIN 51 views

In this article, we will delve into the process of solving a linear equation. Linear equations are fundamental in mathematics and have wide-ranging applications in various fields. Specifically, we will focus on the equation (3n + 3)/5 = (5n - 1)/9, and determine the value of n that satisfies the equation. This type of problem often appears in algebra courses and standardized tests, so mastering the solution techniques is crucial. By understanding the steps involved, you'll gain confidence in solving similar algebraic problems. Our focus will be on providing a clear, step-by-step explanation to ensure that the solution is easy to follow and understand. The objective is to find the precise value of the variable n that makes the equation true. Through careful algebraic manipulation, we will isolate n and arrive at the correct answer. This involves applying the principles of cross-multiplication, distribution, and combining like terms to simplify the equation. This comprehensive approach will empower you to tackle similar algebraic challenges with accuracy and efficiency.

Understanding the Equation

The given equation is (3n + 3)/5 = (5n - 1)/9. This is a linear equation in one variable, n. To solve it, our goal is to isolate n on one side of the equation. This involves several steps, including eliminating the fractions, distributing terms, and combining like terms. The initial structure of the equation presents fractions, which we will address by cross-multiplication. This is a standard technique for eliminating fractions in an equation. Understanding the structure is vital, as it dictates the method we employ to simplify the equation effectively. The left side of the equation consists of the expression (3n + 3) divided by 5, while the right side consists of (5n - 1) divided by 9. Our aim is to find a value for n that makes both sides of the equation equal. This requires a systematic approach to ensure we don't alter the equation's balance during the solution process. By following each step diligently, we can arrive at the correct value of n. It's essential to keep track of each operation performed on both sides of the equation to maintain equality. Let's begin by eliminating the fractions through cross-multiplication.

Step-by-Step Solution

1. Cross-Multiplication

To eliminate the fractions, we will use cross-multiplication. This involves multiplying the numerator of the first fraction by the denominator of the second fraction, and vice versa. Starting with the original equation (3n + 3)/5 = (5n - 1)/9, we multiply (3n + 3) by 9 and (5n - 1) by 5. This gives us: 9 * (3n + 3) = 5 * (5n - 1). This step is crucial as it transforms the equation from a fractional form to a more manageable linear form. Cross-multiplication is based on the principle that if a/b = c/d, then ad = bc. This method ensures we maintain the equation's balance while removing the fractions. Now, our next step is to distribute the multiplication on both sides of the equation. This will involve multiplying the constants outside the parentheses by the terms inside the parentheses. By eliminating the fractions, we have set the stage for further simplification and eventual isolation of the variable n. The resulting equation will allow us to combine like terms and solve for n. This is a standard algebraic technique used to simplify equations and make them easier to solve. Let's proceed to the next step, which involves distributing the terms on both sides of the equation.

2. Distribute Terms

Now, let's distribute the constants on both sides of the equation. We have 9 * (3n + 3) = 5 * (5n - 1). Distributing the 9 on the left side, we get 9 * 3n + 9 * 3, which simplifies to 27n + 27. On the right side, distributing the 5, we get 5 * 5n - 5 * 1, which simplifies to 25n - 5. So, our equation now becomes 27n + 27 = 25n - 5. This step is important because it eliminates the parentheses and allows us to combine like terms in the next step. Distribution involves multiplying the term outside the parentheses by each term inside the parentheses. This is a fundamental algebraic operation that helps in simplifying complex expressions. By distributing terms correctly, we ensure that the equation remains balanced and that we are progressing towards isolating the variable n. The distributed equation now presents us with an opportunity to gather n terms on one side and constant terms on the other. This is the next logical step in solving for n. Let's move on to combining like terms to further simplify the equation.

3. Combine Like Terms

Our equation is currently 27n + 27 = 25n - 5. To combine like terms, we want to get all the n terms on one side and all the constant terms on the other side. Let's subtract 25n from both sides: 27n - 25n + 27 = 25n - 25n - 5, which simplifies to 2n + 27 = -5. Next, we subtract 27 from both sides: 2n + 27 - 27 = -5 - 27, which simplifies to 2n = -32. This step is critical as it brings us closer to isolating n. Combining like terms involves performing the same operation on both sides of the equation to maintain balance. Subtracting 25n from both sides allows us to consolidate the n terms on the left side. Similarly, subtracting 27 from both sides moves all the constant terms to the right side. The result, 2n = -32, is a much simpler equation than what we started with. Now, we have a straightforward equation where we can easily solve for n by dividing both sides by the coefficient of n. Let's proceed to the final step, which is isolating n.

4. Isolate n

We now have the equation 2n = -32. To isolate n, we need to divide both sides of the equation by the coefficient of n, which is 2. Dividing both sides by 2, we get: (2n)/2 = -32/2, which simplifies to n = -16. Therefore, the value of n that makes the original equation true is -16. This final step is essential to obtaining the solution. Isolating n involves performing the inverse operation of whatever operation is being applied to n. In this case, since n is multiplied by 2, we divide both sides by 2 to undo the multiplication. This process reveals the value of n that satisfies the equation. The solution, n = -16, means that if we substitute -16 for n in the original equation, both sides of the equation will be equal. This completes the solution process. We have successfully found the value of n that makes the equation true. Let's verify this solution in the next section to ensure our answer is correct.

Verification

To verify our solution, we substitute n = -16 back into the original equation: (3n + 3)/5 = (5n - 1)/9. Substitute n = -16: (3 * (-16) + 3) / 5 = (5 * (-16) - 1) / 9. Simplify the left side: (3 * (-16) + 3) / 5 = (-48 + 3) / 5 = -45 / 5 = -9. Simplify the right side: (5 * (-16) - 1) / 9 = (-80 - 1) / 9 = -81 / 9 = -9. Since both sides equal -9, our solution n = -16 is correct. Verification is a crucial step in solving equations. It confirms that the value we found for the variable n indeed satisfies the original equation. By substituting the solution back into the equation, we can check if both sides of the equation are equal. If they are, our solution is correct; if not, we need to review our steps to identify any errors. In this case, substituting n = -16 into the original equation resulted in both sides being equal to -9, thus validating our solution. This step ensures accuracy and provides confidence in our answer. Now that we have verified our solution, we can confidently state that the value of n that makes the equation true is -16.

Conclusion

In this article, we have successfully solved the equation (3n + 3)/5 = (5n - 1)/9 and found the value of n to be -16. We followed a step-by-step approach, starting with cross-multiplication to eliminate fractions, distributing terms, combining like terms, and finally isolating n. We then verified our solution by substituting it back into the original equation. Mastering these algebraic techniques is essential for solving linear equations. Solving linear equations is a fundamental skill in algebra, and it has applications in numerous mathematical and real-world contexts. The process we followed—cross-multiplication, distribution, combining like terms, and isolating the variable—is a standard method for solving such equations. By understanding and practicing these steps, you can confidently tackle similar problems. The ability to solve equations like this is not only important for academic purposes but also for problem-solving in everyday life. Understanding the steps involved in solving equations empowers you to approach mathematical challenges with confidence and accuracy. With consistent practice, you can further enhance your skills and apply them to more complex problems.

  • Solving linear equations
  • Cross-multiplication
  • Distributing terms
  • Combining like terms
  • Isolate variables
  • Algebraic equations
  • Equation verification
  • Step-by-step solution
  • Mathematics tutorial
  • Algebra problems