Solving For X In -5/8 X = -160 A Step-by-Step Guide

Introduction

In the realm of mathematics, solving equations is a fundamental skill. Equations are mathematical statements that assert the equality of two expressions. Finding the value of an unknown variable, often denoted as 'x', is a common task. In this article, we will embark on a journey to solve the equation -5/8 x = -160, meticulously exploring each step to unveil the value of 'x'. This equation falls under the category of linear equations, which are algebraic equations where the highest power of the variable is 1. Mastering the techniques to solve such equations is crucial for various mathematical and scientific applications.

Deconstructing the Equation: $-\frac{5}{8} x = -160$

At the heart of our exploration lies the equation $-\frac{5}{8} x = -160$. This equation presents a relationship between the variable 'x' and a numerical value. To decipher the value of 'x', we must isolate it on one side of the equation. This involves employing algebraic manipulations that maintain the equality of both sides. Understanding the structure of the equation is the first step towards unraveling its solution. The equation states that negative five-eighths times 'x' is equal to negative 160. Our goal is to perform operations that undo the multiplication by -5/8, effectively isolating 'x'. This process will lead us to the solution, revealing the numerical value that satisfies the equation.

Step-by-Step Solution: Isolating 'x'

To isolate 'x' in the equation $-\frac{5}{8} x = -160$, we need to counteract the multiplication by -5/8. The key to achieving this is to multiply both sides of the equation by the reciprocal of -5/8. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. Therefore, the reciprocal of -5/8 is -8/5. Multiplying both sides of the equation by -8/5 is a valid algebraic operation because it preserves the equality. This step is crucial as it sets the stage for simplifying the equation and isolating 'x'.

Multiplying by the Reciprocal

Multiplying both sides of the equation $-\frac{5}{8} x = -160$ by -8/5, we get:

(-\frac{8}{5}) * (-\frac{5}{8} x) = (-160) * (-\frac{8}{5})

On the left side, the fractions -5/8 and -8/5 cancel each other out, leaving us with just 'x'. This is because multiplying a fraction by its reciprocal always results in 1. On the right side, we have the product of two negative numbers, which results in a positive number. This step is a pivotal moment in solving the equation, as it simplifies the expression and brings us closer to the value of 'x'.

Simplifying the Equation

After multiplying by the reciprocal, the equation transforms into:

x = (-160) * (-\frac{8}{5})

Now, we need to perform the multiplication on the right side. To do this, we can rewrite -160 as a fraction by placing it over 1: -160/1. Then, we multiply the numerators and the denominators:

x = (-\frac{160}{1}) * (-\frac{8}{5})

x = \frac{(-160) * (-8)}{1 * 5}

x = \frac{1280}{5}

This simplification step involves basic arithmetic operations, ensuring that we maintain the accuracy of the solution. The resulting fraction, 1280/5, represents the value of 'x'. To obtain the final numerical value, we need to perform the division.

Final Calculation: Unveiling the Value of 'x'

To find the value of 'x', we divide 1280 by 5:

x = \frac{1280}{5} = 256

Therefore, the value of 'x' that satisfies the equation $-\frac{5}{8} x = -160$ is 256. This final calculation is the culmination of all the previous steps, revealing the solution to the equation. The value of 'x' is a positive number, indicating that it is located to the right of zero on the number line.

Verifying the Solution: Ensuring Accuracy

To ensure the accuracy of our solution, we can substitute the value of 'x' back into the original equation. This process is known as verification and is a crucial step in problem-solving. By plugging in 'x' = 256 into the equation $-\frac{5}{8} x = -160$, we can check if the equation holds true.

Substituting 'x' = 256

Substituting 'x' = 256 into the original equation, we get:

-\frac{5}{8} * (256) = -160

To simplify the left side, we can multiply the fraction by 256:

-\frac{5 * 256}{8} = -160

-\frac{1280}{8} = -160

-160 = -160

The equation holds true, confirming that our solution 'x' = 256 is correct. This verification step provides confidence in our answer and demonstrates the validity of the solution process.

Conclusion: The Value of 'x' Revealed

In this comprehensive exploration, we have successfully solved the equation $-\frac{5}{8} x = -160$. By meticulously following algebraic principles, we isolated 'x' and determined its value to be 256. The step-by-step solution involved multiplying both sides of the equation by the reciprocal of -5/8, simplifying the equation, and performing the final calculation. To ensure accuracy, we verified our solution by substituting 'x' = 256 back into the original equation. This process not only confirms the correctness of the answer but also reinforces the understanding of equation-solving techniques. Mastering these techniques is essential for success in mathematics and related fields.

Answering the Question: What is the value of $x$ in the equation $-\frac{5}{8} x=-160$?

The value of $x$ in the equation $-\frac{5}{8} x = -160$ is 256.

Therefore, the correct answer is D. 256.

This problem exemplifies the fundamental principles of solving linear equations. By applying algebraic manipulations and verifying the solution, we can confidently determine the value of unknown variables. Such problem-solving skills are invaluable in various academic and real-world scenarios.