Solving -3/4 + (1/8)z = -2/3 A Step-by-Step Guide

Introduction

In this article, we will delve into the step-by-step process of solving the linear equation -3/4 + (1/8)z = -2/3. This equation involves a variable, 'z', and our goal is to isolate 'z' to find its value. Linear equations like this are fundamental in algebra and have applications in various fields, including physics, engineering, and economics. Understanding how to solve them is a crucial skill for anyone studying mathematics or related disciplines. We'll break down each step, explaining the reasoning behind it to ensure a clear and comprehensive understanding of the solution process. By the end of this guide, you'll not only know the answer but also grasp the underlying principles that allow you to solve similar equations with confidence.

Understanding the Equation

Before we dive into the solution, let's break down the equation -3/4 + (1/8)z = -2/3 to make sure we understand each component. The equation consists of several terms: a constant term (-3/4), a term with the variable 'z' ((1/8)z), and another constant term (-2/3). The equal sign (=) indicates that the expression on the left side has the same value as the expression on the right side. Our objective is to find the value of 'z' that makes this equality true. To do this, we'll use algebraic manipulations, which involve performing the same operations on both sides of the equation to maintain the balance and isolate the variable 'z'. These operations include addition, subtraction, multiplication, and division. By carefully applying these operations, we can systematically eliminate terms until we have 'z' by itself on one side of the equation, revealing its value. This process is a cornerstone of algebra and a skill that will be useful in more complex mathematical problems.

Step 1: Isolating the Term with 'z'

Our first step in solving the equation -3/4 + (1/8)z = -2/3 is to isolate the term containing the variable 'z'. This means we want to get the term (1/8)z by itself on one side of the equation. To do this, we need to eliminate the constant term -3/4 from the left side. We can achieve this by adding 3/4 to both sides of the equation. This is a fundamental principle in algebra: whatever operation we perform on one side of the equation, we must perform on the other side to maintain the equality. Adding 3/4 to both sides gives us: -3/4 + 3/4 + (1/8)z = -2/3 + 3/4. On the left side, -3/4 and +3/4 cancel each other out, leaving us with just (1/8)z. On the right side, we need to add the fractions -2/3 and 3/4. To do this, we need a common denominator, which in this case is 12. Converting the fractions, we get -8/12 + 9/12. Adding these gives us 1/12. So, our equation now looks like: (1/8)z = 1/12. This step is crucial because it simplifies the equation and brings us closer to isolating 'z'.

Step 2: Solving for 'z'

Now that we have the equation (1/8)z = 1/12, our next step is to solve for 'z'. The term (1/8)z means 'z' is being multiplied by 1/8. To isolate 'z', we need to perform the inverse operation, which is multiplication by the reciprocal of 1/8. The reciprocal of 1/8 is 8/1, or simply 8. So, we multiply both sides of the equation by 8. This gives us 8 * (1/8)z = 8 * (1/12). On the left side, 8 multiplied by 1/8 cancels out, leaving us with just 'z'. On the right side, we multiply 8 by 1/12. This can be written as 8/1 * 1/12, which simplifies to 8/12. We can further simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 4. Dividing 8 by 4 gives us 2, and dividing 12 by 4 gives us 3. So, 8/12 simplifies to 2/3. Therefore, our solution is z = 2/3. This is the value of 'z' that makes the original equation true. We have successfully isolated 'z' and found its value, completing the solution process.

Verification

To ensure that our solution is correct, it's always a good practice to verify it. We can do this by substituting the value we found for 'z' back into the original equation: -3/4 + (1/8)z = -2/3. We found that z = 2/3, so let's substitute that into the equation: -3/4 + (1/8)(2/3) = -2/3. Now, we need to simplify the left side of the equation. First, we multiply 1/8 by 2/3. This gives us 2/24, which can be simplified to 1/12. So, the equation becomes: -3/4 + 1/12 = -2/3. Next, we need to add -3/4 and 1/12. To do this, we need a common denominator, which is 12. Converting -3/4 to have a denominator of 12, we multiply both the numerator and denominator by 3, giving us -9/12. So, the equation now looks like: -9/12 + 1/12 = -2/3. Adding -9/12 and 1/12 gives us -8/12. This fraction can be simplified by dividing both the numerator and denominator by their greatest common divisor, which is 4. Dividing -8 by 4 gives us -2, and dividing 12 by 4 gives us 3. So, -8/12 simplifies to -2/3. Therefore, the left side of the equation simplifies to -2/3, which is equal to the right side of the equation. This confirms that our solution, z = 2/3, is correct. Verification is a crucial step in problem-solving as it helps catch any potential errors and ensures the accuracy of the final answer.

Conclusion

In this comprehensive guide, we have successfully solved the linear equation -3/4 + (1/8)z = -2/3. We began by understanding the equation and its components. Then, we systematically isolated the term containing the variable 'z' by adding 3/4 to both sides of the equation. This simplified the equation to (1/8)z = 1/12. Next, we solved for 'z' by multiplying both sides of the equation by 8, which is the reciprocal of 1/8. This gave us the solution z = 2/3. To ensure the accuracy of our solution, we verified it by substituting z = 2/3 back into the original equation. The left side of the equation simplified to -2/3, which is equal to the right side, confirming that our solution is correct. Solving linear equations like this is a fundamental skill in mathematics. The steps we've outlined – isolating the variable term and then solving for the variable – can be applied to a wide range of similar problems. By understanding and practicing these steps, you can build confidence in your algebraic abilities and tackle more complex mathematical challenges. Remember, the key to success in mathematics is not just finding the answer but also understanding the process and reasoning behind it.

This detailed walkthrough provides a solid foundation for understanding and solving linear equations. Keep practicing, and you'll find yourself mastering these concepts in no time!