Evaluate The Expression When A=-2, B=3, And C=-6

This article provides a detailed, step-by-step solution to evaluate the given algebraic expression. We will break down each step, ensuring clarity and understanding for readers of all backgrounds. Algebraic expressions are the building blocks of mathematics, and mastering their evaluation is crucial for success in higher-level concepts. This comprehensive guide will not only provide the solution but also enhance your understanding of the fundamental principles involved in simplifying and evaluating such expressions.

Understanding the Expression

Before diving into the solution, it's essential to understand the expression we are working with. The expression is given as: $\frac{2b+2b}{-bc} \frac{36}{36}$. We are given the values of the variables: $a = -2$, $b = 3$, and $c = -6$. Notice that the variable 'a' is provided, but it does not appear in the expression. This means the value of 'a' is irrelevant to the final result. The core of the problem lies in substituting the values of 'b' and 'c' into the expression and simplifying it correctly. When evaluating expressions, we follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures that we perform operations in the correct sequence, leading to the accurate answer. The expression involves multiplication, division, addition, and substitution, all of which we will address in a systematic manner. A firm grasp of these operations is crucial for solving algebraic problems. By understanding the structure of the expression and the given values, we can approach the problem with confidence and accuracy.

Step 1: Simplify the Numerator

The first step in evaluating the expression $\frac{2b+2b}{-bc} \frac{36}{36}$ is to simplify the numerator. The numerator of the fraction is $2b + 2b$. Here, we can combine the like terms, which are both terms containing the variable 'b'. When we add $2b$ and $2b$, we get $4b$. This simplification makes the expression easier to work with in the subsequent steps. Combining like terms is a fundamental algebraic technique that streamlines expressions and reduces the chances of errors. This step sets the stage for the next substitution and simplification operations. By reducing the numerator to its simplest form, we make the entire expression more manageable and accessible. This practice of simplifying expressions early in the process is a key strategy in solving mathematical problems efficiently and accurately. It's also important to note that combining like terms is an application of the distributive property in reverse. For instance, $2b + 2b$ can be seen as $(2+2)b$, which simplifies to $4b$.

Step 2: Simplify the Denominator

Now, let's focus on simplifying the denominator of the fraction $\frac{4b}{-bc} \frac{36}{36}$. The denominator is given as $-bc$. Here, we have a product of three terms: $-1$, $b$, and $c$. To simplify this, we will substitute the given values of $b$ and $c$. We know that $b = 3$ and $c = -6$. Substituting these values into the denominator, we get $-1 * (3) * (-6)$. Multiplying these values together, we first multiply 3 and -6, which gives us -18. Then, multiplying -1 by -18, we get 18. Therefore, the simplified denominator is 18. This process of substituting values and performing the multiplication is a fundamental skill in algebra. It's important to pay close attention to the signs (positive or negative) when multiplying, as this can significantly impact the final result. In this case, multiplying two negative numbers results in a positive number. Simplifying the denominator in this way allows us to proceed with the evaluation of the entire expression with greater ease and accuracy. This step demonstrates the importance of careful substitution and the correct application of multiplication rules.

Step 3: Substitute the value of b into the Simplified Numerator

Having simplified the numerator to $4b$, the next step is to substitute the value of $b$ into this expression. We are given that $b = 3$. Substituting this value, we get $4 * 3$, which equals 12. So, the numerical value of the simplified numerator is 12. This substitution is a straightforward application of algebraic principles. By replacing the variable with its given value, we convert an algebraic term into a numerical value. This process is essential for evaluating expressions and solving equations. The accuracy of this step is crucial, as it directly impacts the final result. It's important to carefully perform the multiplication to avoid errors. This step bridges the gap between the symbolic representation of the expression and its numerical evaluation, bringing us closer to the final solution. The substitution process is a cornerstone of algebraic manipulation, and mastering it is key to success in mathematics.

Step 4: Simplify the Fraction $\frac{36}{36}$

Before we proceed with substituting the values into the main expression, let's simplify the fraction $\frac{36}{36}$. Any non-zero number divided by itself equals 1. Therefore, $\frac{36}{36}$ simplifies to 1. This simplification might seem trivial, but it is an important step in making the overall expression easier to manage. Recognizing and simplifying such fractions is a fundamental skill in arithmetic and algebra. It helps to reduce the complexity of the expression and minimizes the chances of calculation errors. Simplifying fractions to their simplest form is a common practice in mathematics, making it easier to work with and interpret the results. By reducing $\frac{36}{36}$ to 1, we've eliminated a potential source of confusion and streamlined the remaining calculations. This step underscores the importance of looking for opportunities to simplify expressions before diving into more complex operations.

Step 5: Substitute the Values and Simplify the Main Expression

Now, we are ready to substitute the simplified values back into the main expression. Recall that we have simplified the numerator $2b + 2b$ to $4b$, which then became 12 when we substituted $b = 3$. The denominator $-bc$ simplified to 18 when we substituted $b = 3$ and $c = -6$. And the fraction $\frac{36}{36}$ simplified to 1. So, the original expression $\frac{2b+2b}{-bc} \frac{36}{36}$ now becomes $\frac{12}{18} * 1$. The next step is to simplify the fraction $\frac{12}{18}$. Both 12 and 18 are divisible by 6. Dividing both the numerator and the denominator by 6, we get $\frac{12 ÷ 6}{18 ÷ 6} = \frac{2}{3}$. Therefore, the expression simplifies to $\frac{2}{3} * 1$. Finally, multiplying $\frac{2}{3}$ by 1, we get $\frac{2}{3}$. This is the final simplified value of the expression. This comprehensive substitution and simplification process demonstrates the power of breaking down a complex problem into smaller, manageable steps. Each simplification builds upon the previous one, leading to the final solution. The ability to strategically simplify expressions is a critical skill in mathematics, allowing for accurate and efficient problem-solving.

Final Answer

Therefore, the value of the expression $\frac{2b+2b}{-bc} \frac{36}{36}$ when $a = -2$, $b = 3$, and $c = -6$ is $\frac{2}{3}$. This final answer is the result of a step-by-step simplification process, involving the substitution of given values and the application of arithmetic operations. The solution highlights the importance of following the order of operations, simplifying expressions, and carefully performing calculations. By breaking down the problem into smaller, manageable steps, we can arrive at the correct answer with confidence. This process not only provides the solution but also reinforces the fundamental concepts of algebra. Understanding and mastering these concepts is crucial for success in mathematics and related fields. The final answer, $\frac{2}{3}$, represents the culmination of our efforts and demonstrates the effectiveness of a systematic approach to problem-solving.