Solving For Y In 16y = 164 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. The goal of solving an equation is to find the value(s) of the unknown variable(s) that make the equation true. In this article, we will delve into the process of solving a simple algebraic equation to find the value of the variable 'y'. We will specifically address the equation 16y = 164, and through a step-by-step approach, we will unveil the solution and understand the underlying principles involved. This problem falls under the category of basic algebra, which is a cornerstone of mathematical education and has wide-ranging applications in various fields. Mastering the techniques to solve such equations is crucial for anyone looking to build a strong foundation in mathematics and related disciplines.
Understanding the Equation: 16y = 164
Before we embark on the solution, let's dissect the equation 16y = 164. This equation is a linear equation in one variable, 'y'. A linear equation is an equation in which the highest power of the variable is 1. The equation essentially states that 16 multiplied by an unknown value 'y' results in 164. Our mission is to isolate 'y' on one side of the equation to determine its value. The number 16 is the coefficient of 'y', indicating how many times 'y' is being taken. The number 164 is the constant term, representing the result of the multiplication. Understanding the components of the equation is the first step towards solving it. We need to perform operations that maintain the equality of the equation while progressively isolating 'y'. This involves using inverse operations, which are operations that undo each other. For example, multiplication and division are inverse operations, as are addition and subtraction. In this case, since 'y' is being multiplied by 16, we will use division to isolate it. This concept of inverse operations is central to solving algebraic equations and is a skill that will be used repeatedly in more complex mathematical problems.
The Solution: Step-by-Step
To solve the equation 16y = 164, we need to isolate 'y' on one side of the equation. This can be achieved by performing the inverse operation of multiplication, which is division. We will divide both sides of the equation by 16. This maintains the equality of the equation, as we are performing the same operation on both sides. The equation now becomes:
(16y) / 16 = 164 / 16
On the left side, 16 divided by 16 equals 1, leaving us with just 'y'. On the right side, we have 164 divided by 16. This division can be performed using long division or a calculator. When we divide 164 by 16, we get 10 with a remainder of 4. This can be expressed as a mixed number, where the whole number is 10 and the fraction is 4/16. We can simplify the fraction 4/16 by dividing both the numerator and the denominator by their greatest common divisor, which is 4. This gives us the simplified fraction 1/4. Therefore, the value of 164/16 is 10 1/4.
So, the equation simplifies to:
y = 10 1/4
This is the solution to the equation. It means that if we substitute 10 1/4 for 'y' in the original equation, the equation will be true. We can verify this by multiplying 16 by 10 1/4, which should equal 164.
Verification: Ensuring the Solution's Accuracy
To ensure the accuracy of our solution, we need to substitute the value we found for 'y' back into the original equation. This process is called verification and is a crucial step in problem-solving. It helps us catch any potential errors we might have made during the solution process. Our original equation is 16y = 164, and our solution is y = 10 1/4. To substitute, we replace 'y' with 10 1/4:
16 * (10 1/4) = 164
Now, we need to perform the multiplication. First, we convert the mixed number 10 1/4 to an improper fraction. To do this, we multiply the whole number (10) by the denominator (4) and add the numerator (1), which gives us 41. The denominator remains the same, so 10 1/4 is equal to 41/4. Now we can rewrite the equation as:
16 * (41/4) = 164
To multiply a whole number by a fraction, we can write the whole number as a fraction with a denominator of 1:
(16/1) * (41/4) = 164
Now, we multiply the numerators and the denominators:
(16 * 41) / (1 * 4) = 164
656 / 4 = 164
Now, we perform the division:
164 = 164
Since both sides of the equation are equal, our solution is correct. This verification step confirms that y = 10 1/4 is indeed the solution to the equation 16y = 164.
Choosing the Correct Option
Now that we have found the value of y to be 10 1/4, we can identify the correct option from the given choices. The options provided are:
A) 10 1/4 B) 32 C) 16 1/4 D) 180
Comparing our solution with the options, it is clear that option A, 10 1/4, matches our calculated value of y. Therefore, option A is the correct answer.
The other options are incorrect. Option B (32), option C (16 1/4), and option D (180) do not satisfy the equation 16y = 164. If we were to substitute these values for y in the equation, the equation would not be true. For example, 16 * 32 is not equal to 164, 16 * (16 1/4) is not equal to 164, and 16 * 180 is certainly not equal to 164. This highlights the importance of careful calculation and verification in solving equations.
Conclusion
In conclusion, we have successfully solved the equation 16y = 164 and found the value of y to be 10 1/4. We achieved this by using the inverse operation of division to isolate y on one side of the equation. We then verified our solution by substituting it back into the original equation and confirming that it holds true. This step-by-step process demonstrates the fundamental principles of solving linear equations in algebra. The ability to solve such equations is a crucial skill in mathematics and has applications in various fields, including science, engineering, and economics. Understanding the concepts of inverse operations, maintaining equality, and verification are key to mastering algebraic problem-solving. By practicing and applying these principles, you can confidently tackle more complex equations and mathematical challenges. Remember, the key to success in mathematics is a solid understanding of the basics and a systematic approach to problem-solving.