Solving (z-1)/10 = (z+1)/12 A Step-by-Step Guide
Solving algebraic equations is a fundamental skill in mathematics, and this article will provide a detailed walkthrough of solving the linear equation (z-1)/10 = (z+1)/12. We will break down each step, offering clear explanations and insights into the underlying principles. Understanding how to solve such equations is crucial for various mathematical applications and problem-solving scenarios.
The linear equation presented, (z-1)/10 = (z+1)/12, involves fractions and requires careful manipulation to isolate the variable 'z'. Our approach will focus on eliminating the fractions first, simplifying the equation, and then using basic algebraic operations to find the value of 'z'. This step-by-step guide aims to make the process accessible to learners of all levels, from those just starting with algebra to those looking to refresh their skills. By the end of this article, you will have a solid understanding of how to solve this specific equation and similar problems.
This comprehensive guide will not only walk you through the solution but also explain the rationale behind each step, making it easier to apply these techniques to other algebraic problems. The ability to solve equations like this is a cornerstone of mathematical literacy, empowering you to tackle more complex problems in various fields, including science, engineering, and finance. So, let's dive in and master the art of solving linear equations!
Step-by-Step Solution
To solve the equation (z-1)/10 = (z+1)/12, we will follow a series of algebraic steps to isolate the variable 'z'. This process involves eliminating fractions, distributing terms, combining like terms, and finally, solving for 'z'. Each step is crucial and builds upon the previous one to arrive at the correct solution. By carefully following these steps, you will gain a clear understanding of how to solve similar equations.
1. Eliminate Fractions
The first step in solving this equation is to eliminate the fractions. This makes the equation easier to work with and reduces the chances of making errors. To do this, we need to find the least common multiple (LCM) of the denominators, which are 10 and 12. The LCM of 10 and 12 is 60. We multiply both sides of the equation by 60 to clear the fractions:
60 * [(z-1)/10] = 60 * [(z+1)/12]
This simplifies to:
6(z-1) = 5(z+1)
By multiplying both sides of the equation by the LCM, we have successfully eliminated the fractions, making the equation more manageable. This technique is fundamental in solving equations involving fractions and is a key step in simplifying the problem.
2. Distribute Terms
Now that we have eliminated the fractions, the next step is to distribute the numbers on both sides of the equation. This involves multiplying the numbers outside the parentheses by each term inside the parentheses. On the left side, we distribute 6 to (z-1), and on the right side, we distribute 5 to (z+1):
6 * z - 6 * 1 = 5 * z + 5 * 1
This simplifies to:
6z - 6 = 5z + 5
Distributing terms is a crucial step in simplifying algebraic expressions and equations. It allows us to remove parentheses and combine like terms, which is necessary for isolating the variable.
3. Combine Like Terms
After distributing the terms, the next step is to combine like terms. This involves grouping terms with the same variable (in this case, 'z') on one side of the equation and constant terms on the other side. To do this, we can subtract 5z from both sides and add 6 to both sides:
6z - 6 - 5z = 5z + 5 - 5z
This simplifies to:
z - 6 = 5
Next, we add 6 to both sides:
z - 6 + 6 = 5 + 6
This simplifies to:
z = 11
Combining like terms is a fundamental skill in algebra. It allows us to simplify equations and isolate the variable, bringing us closer to the solution. This step often involves performing the same operation on both sides of the equation to maintain balance.
4. Solve for z
After combining like terms, we have simplified the equation to a point where we can easily solve for 'z'. From the previous step, we have:
z = 11
This means that the value of 'z' that satisfies the original equation is 11. To verify this, we can substitute z = 11 back into the original equation and check if both sides are equal.
(11 - 1)/10 = (11 + 1)/12
10/10 = 12/12
1 = 1
Since both sides are equal, our solution is correct. Solving for the variable is the ultimate goal in solving an equation. It involves isolating the variable on one side of the equation, which gives us its value.
Verification of the Solution
To ensure the accuracy of our solution, it's essential to verify it by substituting the value of 'z' back into the original equation. This step confirms that our algebraic manipulations were correct and that the value we found for 'z' indeed satisfies the equation. The verification process involves replacing 'z' with 11 in the original equation and simplifying both sides to see if they are equal.
Substituting z = 11
Let's substitute z = 11 into the original equation:
(z - 1)/10 = (z + 1)/12
Replace 'z' with 11:
(11 - 1)/10 = (11 + 1)/12
Now, simplify both sides of the equation.
Simplifying Both Sides
First, simplify the left side:
(11 - 1)/10 = 10/10 = 1
Next, simplify the right side:
(11 + 1)/12 = 12/12 = 1
Since both sides simplify to 1, the equation holds true:
1 = 1
This confirms that z = 11 is the correct solution to the equation (z - 1)/10 = (z + 1)/12. Verification is a crucial step in problem-solving, as it provides assurance that the solution is accurate and that no errors were made during the process.
Common Mistakes to Avoid
Solving equations can be tricky, and there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid errors and solve equations more accurately. Some common mistakes include incorrect distribution, errors in combining like terms, and mistakes in arithmetic. Let's look at each of these in detail.
1. Incorrect Distribution
One of the most frequent errors occurs during the distribution step. This involves multiplying a number outside parentheses by each term inside the parentheses. A common mistake is to forget to multiply the number by every term. For example, in the equation 6(z - 1) = 5(z + 1), incorrectly distributing the 6 might lead to 6z - 1 instead of the correct 6z - 6. To avoid this, always double-check that you have multiplied the term outside the parentheses by each term inside.
2. Errors in Combining Like Terms
Another common mistake is combining like terms incorrectly. This often happens when students forget the signs (positive or negative) or when they try to combine terms that are not alike. For instance, in the equation 6z - 6 = 5z + 5, a mistake might be made by adding 6 to 5z instead of adding it to the constant term 5. To avoid these errors, carefully identify like terms and pay close attention to the signs before combining them.
3. Mistakes in Arithmetic
Arithmetic errors, such as incorrect addition, subtraction, multiplication, or division, can also lead to wrong solutions. Even a small arithmetic error can throw off the entire process. For example, if you incorrectly add 5 + 6 and get 10 instead of 11, your final answer will be wrong. To minimize arithmetic errors, it's helpful to double-check your calculations and use a calculator if necessary. Taking your time and working methodically can also reduce the likelihood of making such mistakes.
Tips to Avoid Mistakes
To avoid these common mistakes, consider the following tips:
- Write each step clearly: Show all your work, so it's easier to spot any errors.
- Double-check your work: Before moving on to the next step, review your calculations.
- Use a calculator: For complex calculations, a calculator can help reduce arithmetic errors.
- Verify your solution: Substitute your answer back into the original equation to check if it is correct.
- Practice regularly: The more you practice, the more comfortable you will become with solving equations, and the fewer mistakes you will make.
Conclusion
In conclusion, solving the equation (z-1)/10 = (z+1)/12 involves several key steps: eliminating fractions, distributing terms, combining like terms, and solving for 'z'. By following these steps carefully and understanding the underlying principles, you can solve similar algebraic equations effectively. The solution to the equation is z = 11, which we verified by substituting it back into the original equation.
Throughout this comprehensive guide, we have emphasized the importance of each step and provided clear explanations to enhance understanding. We also highlighted common mistakes to avoid, such as incorrect distribution, errors in combining like terms, and arithmetic errors. By being aware of these pitfalls and following our tips, you can improve your accuracy and confidence in solving algebraic equations.
Mastering algebraic equations is a fundamental skill in mathematics, with applications in various fields, including science, engineering, and finance. The ability to solve these equations empowers you to tackle more complex problems and make informed decisions. We encourage you to continue practicing and applying these techniques to further develop your mathematical skills. Remember, the key to success in mathematics is consistent practice and a thorough understanding of the fundamental concepts.
This article has provided you with a detailed walkthrough of solving a specific linear equation. However, the principles and techniques discussed here can be applied to a wide range of algebraic problems. Keep practicing, stay curious, and continue to explore the fascinating world of mathematics!
