Evaluate 6y² + 3y - ¼ For Y = 1/6

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This article will guide you through the process of evaluating algebraic expressions, specifically focusing on the order of operations when dealing with fractional values. We will tackle the problem of evaluating 6y² + 3y - ¼ for y = 1/6. This involves understanding how to substitute values into an expression and then apply the correct mathematical rules to arrive at the correct answer. By the end of this discussion, you'll be confident in handling similar problems involving algebraic expressions and fractions.

Understanding the Order of Operations (PEMDAS/BODMAS)

The first crucial step in evaluating any mathematical expression is to understand and correctly apply the order of operations. This set of rules ensures that everyone arrives at the same answer for a given problem. The most common acronyms for remembering this order are PEMDAS (Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right)) and BODMAS (Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right)). Let's break down what each part means:

  • Parentheses / Brackets: Operations inside parentheses or brackets are always performed first.
  • Exponents / Orders: Next, we handle exponents or powers (like squaring a number).
  • Multiplication and Division: These operations are performed next, working from left to right as they appear in the expression.
  • Addition and Subtraction: Finally, addition and subtraction are performed, also from left to right.

Mastering PEMDAS/BODMAS is fundamental not only for this specific problem but for all of mathematics. It provides a consistent framework for simplifying complex expressions and ensuring accuracy in calculations. Without a standardized order, mathematical expressions could yield vastly different results depending on how they were interpreted, leading to confusion and errors. Therefore, when we approach 6y² + 3y - ¼, we must keep this hierarchy of operations firmly in mind to avoid missteps.

Substituting the Value of y

Our expression is 6y² + 3y - ¼, and we are given that y = 1/6. The next step is to substitute this value of y into the expression wherever y appears. It's crucial to use parentheses when substituting a fraction or a negative number to avoid errors, especially when dealing with exponents.

So, 6y² becomes 6 * (1/6)². And 3y becomes 3 * (1/6).

The expression now looks like this: 6 * (1/6)² + 3 * (1/6) - ¼.

This substitution is a critical phase because any mistake made here will propagate through the entire calculation. It’s a good practice to write out the expression with the substituted value clearly, perhaps even double-checking it to ensure each instance of y has been correctly replaced by (1/6). This meticulous approach ensures that the foundation of our calculation is sound before we move on to applying the order of operations.

Evaluating the Exponent

Following PEMDAS/BODMAS, the next step after substitution is to handle the exponents. In our expression, we have (1/6)². Squaring a fraction means multiplying the fraction by itself.

So, (1/6)² = (1/6) * (1/6).

To multiply fractions, we multiply the numerators together and the denominators together:

(1 * 1) / (6 * 6) = 1/36.

Now, we substitute this value back into our expression:

6 * (1/36) + 3 * (1/6) - ¼.

This step is where the fractional arithmetic starts to come into play. Squaring a fraction might seem straightforward, but it's essential to remember the rule for multiplying fractions. The result of the exponentiation, 1/36, is now ready to be used in the next phase of our calculation. This careful progression, adhering to the order of operations, ensures that we are systematically simplifying the expression. Each part, from substitution to exponentiation, builds upon the previous one, guiding us toward the final answer.

Performing Multiplication and Division

With the exponent evaluated, we now move to the multiplication and division step, remembering to perform these operations from left to right. Our expression is currently 6 * (1/36) + 3 * (1/6) - ¼.

First, let's tackle 6 * (1/36). To multiply a whole number by a fraction, we can write the whole number as a fraction with a denominator of 1:

6/1 * 1/36.

Now, multiply the numerators and the denominators:

(6 * 1) / (1 * 36) = 6/36.

This fraction can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 6:

6 ÷ 6 / 36 ÷ 6 = 1/6.

Next, let's evaluate 3 * (1/6). Again, write the whole number as a fraction:

3/1 * 1/6.

Multiply the numerators and denominators:

(3 * 1) / (1 * 6) = 3/6.

Simplify this fraction by dividing both the numerator and the denominator by 3:

3 ÷ 3 / 6 ÷ 3 = 1/2.

Our expression now becomes: 1/6 + 1/2 - ¼.

This phase involves careful fraction multiplication and simplification. Each multiplication step, 6 * (1/36) and 3 * (1/6), needs to be handled precisely. Recognizing the opportunity to simplify fractions like 6/36 to 1/6 and 3/6 to 1/2 is key to making the subsequent addition and subtraction steps more manageable. This process demonstrates how simplifying intermediate results can streamline the overall calculation, reducing the complexity of the numbers we have to work with.

Conducting Addition and Subtraction

The final step in our evaluation is addition and subtraction, working from left to right. Our expression is now 1/6 + 1/2 - ¼.

First, let's perform the addition: 1/6 + 1/2. To add fractions, they must have a common denominator. The least common multiple (LCM) of 6 and 2 is 6. So, we need to convert 1/2 to an equivalent fraction with a denominator of 6.

Multiply the numerator and denominator of 1/2 by 3:

(1 * 3) / (2 * 3) = 3/6.

Now, we can add the fractions:

1/6 + 3/6 = (1 + 3) / 6 = 4/6.

This fraction can be simplified by dividing the numerator and denominator by 2:

4 ÷ 2 / 6 ÷ 2 = 2/3.

Now, our expression is 2/3 - ¼.

Next, we perform the subtraction: 2/3 - ¼. Again, we need a common denominator. The LCM of 3 and 4 is 12.

Convert 2/3 to an equivalent fraction with a denominator of 12: Multiply the numerator and denominator by 4:

(2 * 4) / (3 * 4) = 8/12.

Convert 1/4 to an equivalent fraction with a denominator of 12: Multiply the numerator and denominator by 3:

(1 * 3) / (4 * 3) = 3/12.

Now, perform the subtraction:

8/12 - 3/12 = (8 - 3) / 12 = 5/12.

The value of the expression 6y² + 3y - ¼ when y = 1/6 is 5/12.

This final stage, focusing on addition and subtraction of fractions, requires a solid understanding of finding common denominators. Converting fractions to equivalent forms with a shared denominator is a cornerstone of fractional arithmetic. The process of finding the LCM and then adjusting the numerators ensures that we are accurately combining or separating these fractional parts. Each step, from finding the common denominator for 1/6 and 1/2 to doing the same for 2/3 and 1/4, is executed systematically. The final answer, 5/12, is the culmination of applying all the rules of the order of operations and fraction manipulation correctly.

Evaluating Another Expression

Let's consider another example to solidify your understanding of evaluating expressions, this time with decimals: (1.584 + 1.408) ÷ 4.

We follow the order of operations (PEMDAS/BODMAS). First, we handle the operations inside the parentheses:

1.584 + 1.408.

Adding these decimals: 1.584

  • 1.408

2.992

So, the expression becomes 2.992 ÷ 4.

Next, we perform the division:

2.992 ÷ 4.

Dividing 2.992 by 4:

   0.748
  ______
4 | 2.992
  - 2 8
  -----
    19
  - 16
  ----
     32
   - 32
   ----
      0

The value of the expression (1.584 + 1.408) ÷ 4 is 0.748.

This example illustrates how the order of operations applies equally well to decimal numbers. The parentheses dictate that the addition must be performed first, and only after that sum is obtained do we proceed with the division. This straightforward application of PEMDAS reinforces the universality of these mathematical rules, regardless of whether you are working with fractions, decimals, or whole numbers. Each step is clear and directly follows the established principles of arithmetic.

Conclusion

Evaluating algebraic expressions, whether they involve fractions or decimals, hinges on a steadfast adherence to the order of operations (PEMDAS/BODMAS). We successfully evaluated 6y² + 3y - ¼ for y = 1/6 by meticulously substituting the value, calculating the exponent, performing multiplications, and finally executing the addition and subtraction, arriving at the answer 5/12. Similarly, for (1.584 + 1.408) ÷ 4, we first added the numbers in parentheses and then divided the sum by 4, resulting in 0.748.

Practice is key to mastering these skills. The more you work through various problems, the more intuitive the application of PEMDAS/BODMAS will become. Remember to break down complex expressions into smaller, manageable steps. Always double-check your substitutions and calculations, especially when dealing with fractions and their common denominators.

For further exploration into the fundamental rules of mathematics and algebraic simplification, you can consult resources like Khan Academy. They offer a wealth of free lessons and practice exercises that can help solidify your understanding of these essential concepts. Additionally, exploring websites dedicated to math education can provide different perspectives and additional problem-solving strategies.