Solving The Equation 16y = 164 A Step By Step Guide

by ADMIN 52 views

In the realm of mathematics, solving equations is a fundamental skill. This article delves into the process of finding the value of 'y' in the equation 16y = 164. We will explore the step-by-step method to isolate 'y' and arrive at the correct solution. Understanding how to solve such equations is crucial for various mathematical applications and problem-solving scenarios.

Breaking Down the Equation

At the heart of our exploration lies the equation 16y = 164. This equation represents a simple algebraic relationship where 'y' is an unknown variable. Our goal is to isolate 'y' on one side of the equation to determine its value. The coefficient '16' is multiplied by 'y', and the result equals '164'. To find 'y', we need to perform the inverse operation of multiplication, which is division.

Isolating the Variable 'y'

To isolate 'y', we must divide both sides of the equation by the coefficient '16'. This maintains the equality of the equation while moving the coefficient away from 'y'. Mathematically, this can be represented as:

(16y) / 16 = 164 / 16

Performing this division on both sides simplifies the equation significantly. The '16' on the left side cancels out, leaving 'y' isolated. On the right side, we perform the division of 164 by 16. This step is crucial in determining the numerical value of 'y'.

Performing the Division

Dividing 164 by 16 yields the result 10.25. This can be expressed as a decimal or as a mixed number. In this case, 10.25 is the decimal representation of the fraction 164/16. We can also express this as a mixed number, which is 10 and 1/4. Understanding both representations is beneficial for different contexts and applications.

Expressing the Solution

The solution to the equation 16y = 164 is y = 10.25 or y = 10 1/4. These two forms represent the same value, with 10.25 being the decimal form and 10 1/4 being the mixed number form. Choosing the appropriate form depends on the specific context or the desired level of precision. Both forms are mathematically correct and equivalent.

Analyzing the Options

Now, let's analyze the provided options in the context of our solution:

  • A. 180: This value is significantly different from our calculated solution of 10.25 or 10 1/4. Therefore, this option is incorrect.
  • B. 16 1/4: This option is close to our solution but not quite the same. It represents 16.25, which is different from 10.25. Thus, this option is also incorrect.
  • C. 32: This value is substantially larger than our solution and is therefore incorrect.
  • D. 10 1/4: This option perfectly matches our calculated solution of y = 10 1/4. This is the correct answer.

Identifying the Correct Answer

Based on our step-by-step solution and analysis of the options, the correct answer is D. 10 1/4. This value satisfies the equation 16y = 164, making it the accurate solution. Understanding the process of solving the equation and comparing it with the provided options is essential for arriving at the correct answer.

Key Takeaways

  • Isolate the variable: The primary goal in solving equations is to isolate the variable on one side of the equation.
  • Perform inverse operations: Use inverse operations (e.g., division for multiplication) to isolate the variable.
  • Maintain equality: Ensure that any operation performed on one side of the equation is also performed on the other side to maintain equality.
  • Express solutions in different forms: Understand how to express solutions as decimals, fractions, and mixed numbers.
  • Analyze options carefully: Compare your solution with the provided options to identify the correct answer.

Expanding on Equation Solving Techniques

Understanding Linear Equations

The equation 16y = 164 is a linear equation, a fundamental concept in algebra. Linear equations involve variables raised to the power of one and can be represented graphically as a straight line. Solving linear equations is a basic skill that forms the foundation for more complex algebraic problems. The key to solving linear equations is to isolate the variable using inverse operations.

The Importance of Inverse Operations

Inverse operations are crucial in equation solving. For example, the inverse operation of addition is subtraction, and the inverse operation of multiplication is division. In the equation 16y = 164, we used division (the inverse of multiplication) to isolate 'y'. Understanding and applying inverse operations correctly is essential for solving a wide range of algebraic equations. This concept is not only applicable in basic algebra but also extends to more advanced mathematical fields.

Checking Your Solution

After solving an equation, it's always a good practice to check your solution. This involves substituting the value you found for the variable back into the original equation. If the equation holds true, then your solution is correct. For example, substituting y = 10 1/4 back into 16y = 164 should result in a true statement. This step ensures accuracy and helps prevent errors.

Applications of Linear Equations

Linear equations have numerous applications in real-world scenarios. They can be used to model relationships between quantities, solve problems involving rates and proportions, and make predictions based on data. For instance, linear equations can be used to calculate the cost of goods, determine the distance traveled at a certain speed, or predict future sales based on past trends. The versatility of linear equations makes them an essential tool in various fields, including science, engineering, economics, and finance.

Conclusion: Mastering the Art of Equation Solving

In conclusion, solving for 'y' in the equation 16y = 164 involves a systematic approach of isolating the variable using inverse operations. The correct solution is y = 10 1/4, which can also be expressed as 10.25. This exercise highlights the fundamental principles of equation solving and their application in mathematical problem-solving. By understanding the concepts of inverse operations, maintaining equality, and expressing solutions in different forms, you can confidently tackle a variety of algebraic equations. Mastering these skills is crucial for success in mathematics and its applications in various fields. The ability to solve equations accurately and efficiently is a valuable asset in both academic and professional settings.

This comprehensive guide has not only provided the solution to the specific equation but has also expanded on the broader concepts of equation solving, emphasizing the importance of understanding the underlying principles and techniques. By practicing these techniques and applying them to different problems, you can enhance your mathematical skills and build a strong foundation for future learning.