Simplifying And Evaluating Algebraic Expressions Step-by-Step Guide

Understanding the Basics of Algebraic Expressions

Before diving into the specifics of our expression, it's crucial to understand what algebraic expressions are and the basic operations involved. An algebraic expression is a combination of variables, constants, and mathematical operations such as addition, subtraction, multiplication, and division. Variables are symbols (usually letters) that represent unknown values, while constants are fixed numerical values. Simplifying an algebraic expression means rewriting it in a more concise and manageable form, typically by combining like terms and reducing the expression to its simplest equivalent. Evaluating an algebraic expression involves substituting specific numerical values for the variables and performing the arithmetic operations to find the numerical value of the expression. This process is fundamental in various mathematical contexts, including solving equations, modeling real-world scenarios, and analyzing mathematical relationships. A solid understanding of these basics is essential for mastering more advanced algebraic concepts and techniques.

Step 1: Simplify the Numerator

The first step in simplifying the expression $\frac{9 t^3+4 t^3-3 t^3}{11 t-5 t}$ is to focus on the numerator, which is $9 t^3+4 t^3-3 t^3$. In this part of the expression, we have three terms, all of which contain the same variable raised to the same power ($t^3$). These are known as like terms, and they can be combined by adding or subtracting their coefficients. The coefficients are the numerical parts of the terms; in this case, they are 9, 4, and -3. To combine these like terms, we simply add the coefficients together: $9 + 4 - 3 = 10$. This means that $9 t^3+4 t^3-3 t^3$ simplifies to $10t^3$. This simplification is a crucial step because it reduces the complexity of the expression, making it easier to work with in subsequent steps. By combining like terms, we are essentially streamlining the expression to its most basic form, which is a key principle in algebraic manipulation.

Step 2: Simplify the Denominator

Next, we turn our attention to the denominator of the expression, which is $11 t-5 t$. Similar to the numerator, the denominator also consists of like terms. Both terms contain the variable $t$ raised to the power of 1. To simplify this, we subtract the coefficients: $11 - 5 = 6$. Therefore, $11 t-5 t$ simplifies to $6t$. This step is analogous to simplifying the numerator; by combining like terms, we reduce the complexity of the denominator, making the entire expression more manageable. Simplifying the denominator is just as important as simplifying the numerator, as it sets the stage for further simplification or evaluation of the expression. The result, $6t$, is a simplified form that allows us to proceed with the next steps in a clear and straightforward manner. This methodical approach of simplifying each part of the expression separately is a fundamental technique in algebra.

Step 3: Simplify the Entire Expression

Now that we have simplified both the numerator and the denominator, our expression looks like this: $\frac{10t^3}{6t}$. To further simplify this fraction, we can look for common factors in both the numerator and the denominator. First, let’s consider the numerical coefficients: 10 and 6. The greatest common factor (GCF) of 10 and 6 is 2. So, we can divide both coefficients by 2, which gives us $\frac{5}{3}$. Next, we consider the variable part of the expression. We have $t^3$ in the numerator and $t$ in the denominator. Using the rules of exponents, when dividing like bases, we subtract the exponents. In this case, we have $t^{3-1} = t^2$. Therefore, the simplified expression is $\frac{5t^2}{3}$. This step of simplifying the entire expression involves both numerical and variable factors, demonstrating a comprehensive understanding of algebraic simplification techniques. By reducing the fraction to its simplest form, we make it easier to work with and understand, which is a key goal in algebra.

Step 4: Substitute the Value of t

The next step is to evaluate the simplified expression when $t=6$. This means we will replace every instance of the variable $t$ in the expression $\frac{5t^2}{3}$ with the number 6. When we substitute $t$ with 6, we get $\frac{5(6)^2}{3}$. Now, we need to follow the order of operations (PEMDAS/BODMAS), which tells us to handle exponents before multiplication and division. So, we first calculate $6^2$, which is $6 \times 6 = 36$. Our expression now looks like this: $\frac{5(36)}{3}$. This substitution step is a critical part of evaluating algebraic expressions, as it allows us to find the numerical value of the expression for a specific value of the variable. The careful substitution ensures that we are setting up the expression correctly for the final calculation. Understanding how to substitute values correctly is fundamental in algebra and is used extensively in solving equations and modeling real-world problems.

Step 5: Evaluate the Expression

Now that we have substituted $t=6$ into the simplified expression, we have $\frac{5(36)}{3}$. The next step is to perform the multiplication in the numerator: $5 \times 36 = 180$. So, the expression becomes $\frac{180}{3}$. Finally, we perform the division: $180 \div 3 = 60$. Therefore, the value of the expression $\frac{9 t^3+4 t^3-3 t^3}{11 t-5 t}$ when $t=6$ is 60. This final calculation completes the evaluation process, providing us with a numerical answer. The steps of multiplication and division are performed in the correct order to ensure the accuracy of the result. This methodical approach to evaluating the expression demonstrates a thorough understanding of algebraic principles and arithmetic operations. The final answer of 60 is the culmination of the simplification and substitution steps, showcasing the power of algebraic manipulation in determining the value of expressions.

Conclusion

In conclusion, we have successfully simplified the algebraic expression $\frac{9 t^3+4 t^3-3 t^3}{11 t-5 t}$ and evaluated it for $t=6$. The process involved several key steps: first, we simplified the numerator and denominator by combining like terms. Then, we simplified the entire expression by canceling common factors. Next, we substituted the value of $t$ into the simplified expression, and finally, we performed the arithmetic operations to find the numerical value. This step-by-step approach not only helps in solving this particular problem but also provides a general method for simplifying and evaluating algebraic expressions. Understanding these principles is crucial for success in algebra and higher-level mathematics. By mastering these techniques, you can confidently tackle similar problems and apply these skills in various mathematical contexts. The ability to simplify and evaluate algebraic expressions is a fundamental skill that opens doors to more advanced mathematical concepts and applications. Remember, practice is key to mastering these skills, so continue to work through examples and challenge yourself with new problems.