Solving -9/8 * A = 45 A Step-by-Step Guide

In the realm of mathematics, solving equations is a fundamental skill. Equations represent relationships between variables and constants, and finding the value(s) of the variable(s) that make the equation true is the core of the solving process. This article delves into the step-by-step solution of a specific algebraic equation: -9/8 * a = 45. We'll break down the process, making it easy to understand for anyone, regardless of their mathematical background. Understanding how to solve such equations is crucial for various applications in science, engineering, economics, and everyday problem-solving. So, let's embark on this mathematical journey and unravel the value of the variable 'a' in this equation. This involves isolating the variable 'a' on one side of the equation, which will reveal its numerical value. We will accomplish this by performing operations on both sides of the equation, ensuring that the equality remains balanced. This fundamental principle of maintaining balance is at the heart of solving any algebraic equation, and mastering this concept will empower you to tackle a wide range of mathematical problems with confidence.

The beauty of algebra lies in its ability to represent real-world situations with mathematical symbols and equations. Once we have an equation, we can use algebraic techniques to solve for unknown quantities, allowing us to make predictions, solve problems, and gain deeper insights into the world around us. Whether you're a student learning algebra for the first time, or someone looking to brush up on your math skills, this article will provide you with a clear and concise guide to solving this specific equation and, more importantly, the underlying principles that you can apply to countless other mathematical challenges. So, let's dive in and unlock the power of algebra!

Before diving into the solution, let's dissect the equation -9/8 * a = 45 to ensure a clear understanding of its components. This equation is a linear equation in one variable, where 'a' represents the unknown quantity we aim to determine. The coefficient of 'a' is -9/8, which is a fraction indicating that 'a' is being multiplied by a negative value less than one. On the right side of the equation, we have the constant 45, which represents the value the expression on the left side must equal. The asterisk symbol (*) represents multiplication. Understanding these basic elements is essential for developing a strategy to solve the equation effectively. We need to isolate 'a' on one side of the equation. This means we need to get rid of the coefficient -9/8. The key to achieving this lies in using inverse operations. Since 'a' is being multiplied by -9/8, we can undo this operation by dividing both sides of the equation by -9/8. Remember, whatever operation we perform on one side of the equation, we must perform on the other side to maintain the balance and ensure the equation remains true.

The concept of inverse operations is a cornerstone of algebra. Addition and subtraction are inverse operations, as are multiplication and division. By applying the appropriate inverse operation, we can systematically isolate the variable we want to solve for. In this case, dividing by -9/8 is the key to unlocking the value of 'a'. As we proceed with the solution, we'll see how this simple yet powerful technique allows us to navigate the equation and arrive at the correct answer. Keep in mind that understanding the structure of the equation and the relationship between its components is crucial for choosing the right approach and avoiding common mistakes. With a solid grasp of these fundamentals, you'll be well-equipped to tackle a wide variety of algebraic challenges.

To isolate 'a' in the equation -9/8 * a = 45, we need to perform the inverse operation of multiplication, which is division. We will divide both sides of the equation by the coefficient of 'a', which is -9/8. This step is crucial for isolating 'a' on one side of the equation. Remember, the principle of maintaining balance in an equation dictates that any operation performed on one side must also be performed on the other side. By dividing both sides by -9/8, we ensure that the equality remains valid and that we are moving closer to finding the value of 'a'. This process is a fundamental technique in algebra and is used extensively to solve various types of equations.

Dividing by a fraction can sometimes seem tricky, but it's simply a matter of multiplying by the reciprocal of the fraction. The reciprocal of -9/8 is -8/9. So, dividing by -9/8 is the same as multiplying by -8/9. This clever trick allows us to transform the division problem into a multiplication problem, which is often easier to handle. By multiplying both sides of the equation by -8/9, we effectively cancel out the -9/8 on the left side, leaving 'a' isolated. This step demonstrates the power of inverse operations and how they can be used to simplify equations and reveal the value of the unknown variable.

Let's perform the operation:

(-9/8 * a) / (-9/8) = 45 / (-9/8)

This simplifies to:

a = 45 * (-8/9)

Now, we multiply 45 by -8/9. To do this, we can think of 45 as 45/1 and then multiply the numerators and the denominators:

a = (45 * -8) / (1 * 9)

a = -360 / 9

Finally, we simplify the fraction by dividing -360 by 9:

a = -40

Therefore, the solution to the equation -9/8 * a = 45 is a = -40. This means that if we substitute -40 for 'a' in the original equation, the equation will hold true. This is a crucial step in verifying our solution and ensuring that we have arrived at the correct answer. The process of solving equations involves careful manipulation and attention to detail. By following the steps outlined above, we can confidently arrive at the solution and gain a deeper understanding of the underlying algebraic principles.

To ensure the accuracy of our solution, it's essential to verify that a = -40 indeed satisfies the original equation, -9/8 * a = 45. Verification is a critical step in the problem-solving process, as it allows us to catch any potential errors and confirm that our answer is correct. By substituting the value we found for 'a' back into the original equation, we can see if both sides of the equation are equal. If they are, then we can be confident that our solution is valid. This process not only confirms our answer but also reinforces our understanding of the equation and the relationship between its components.

Let's substitute a = -40 into the equation:

-9/8 * (-40) = 45

We can simplify the left side of the equation by multiplying -9/8 by -40. Remember that multiplying two negative numbers results in a positive number:

(-9/8) * (-40/1) = (9 * 40) / (8 * 1)

= 360 / 8

Now, we simplify the fraction 360/8:

360 / 8 = 45

So, the left side of the equation simplifies to 45, which is equal to the right side of the equation. This confirms that our solution, a = -40, is correct. The verification process is a powerful tool for building confidence in our mathematical abilities. By taking the time to check our work, we can ensure that we are not only arriving at the correct answers but also developing a deeper understanding of the concepts involved. This habit of verification is crucial for success in mathematics and in any field that requires problem-solving skills.

In this article, we successfully solved the equation -9/8 * a = 45 and determined that a = -40. We accomplished this by applying the fundamental principles of algebra, including the use of inverse operations and the importance of maintaining balance in an equation. The step-by-step solution demonstrated the process of isolating the variable 'a' and arriving at its numerical value. We also emphasized the critical role of verification in ensuring the accuracy of our solution. This process of solving equations is not just a mathematical exercise; it's a fundamental skill that has wide-ranging applications in various fields, from science and engineering to finance and everyday problem-solving. By mastering these techniques, you can unlock the power of algebra and apply it to a multitude of real-world scenarios.

The ability to solve equations is a cornerstone of mathematical literacy. It empowers us to make sense of the world around us, to analyze data, and to make informed decisions. Whether you're balancing your budget, calculating the trajectory of a projectile, or modeling complex systems, the skills you learn in algebra will serve you well. This article has provided a clear and concise guide to solving a specific equation, but the principles and techniques we've discussed are applicable to a wide range of algebraic problems. So, continue practicing, exploring, and challenging yourself, and you'll find that your mathematical abilities will continue to grow and expand. Remember, the journey of learning mathematics is a continuous one, and every equation solved is a step forward.

Solving equations, algebra, linear equations, inverse operations, verification, mathematics, problem-solving, coefficient, variable, constant.