Evaluate (st - 17s - T) / S For S=-5, T=5 | Step-by-Step
Welcome! In this article, we'll walk through the process of evaluating the expression (st - 17s - t) / s when s = -5 and t = 5. This involves substituting the given values into the expression and simplifying the result. If you're tackling algebra problems or just brushing up on your math skills, you're in the right place. Let's dive in and break it down step by step.
Understanding the Expression
Before we start substituting, let’s take a closer look at the expression: (st - 17s - t) / s. This algebraic expression involves variables s and t, and it includes multiplication, subtraction, and division. To evaluate this expression, we'll replace s and t with their given values and follow the order of operations (PEMDAS/BODMAS) to simplify. This means dealing with parentheses, exponents, multiplication and division (from left to right), and addition and subtraction (from left to right).
Breaking Down the Components
- st: This means s multiplied by t. It’s a crucial component as it sets the stage for the initial calculation.
- 17s: This term represents 17 times s, highlighting the impact of the variable s on the overall expression.
- t: The variable t stands alone, but its value is equally important in the subtraction process.
- / s: The entire expression is divided by s, which means the value of s plays a significant role in the final outcome. If s were zero, the expression would be undefined.
By understanding these components, we can better approach the substitution and simplification process. Each term contributes to the final result, and handling them in the correct order is key to accuracy.
Step 1: Substitute the Values
The first step in evaluating the expression is to substitute the given values for s and t. We are given that s = -5 and t = 5. Let's replace these values into the expression:
(st - 17s - t) / s becomes ((-5)(5) - 17(-5) - 5) / (-5)
This substitution transforms the algebraic expression into a numerical one, which we can then simplify using the order of operations. It’s crucial to substitute correctly, paying attention to signs and ensuring each variable is replaced accurately. This step is fundamental as any error here will propagate through the rest of the solution.
Why Substitution Matters
Substitution is a fundamental concept in algebra. It allows us to take abstract expressions and turn them into concrete numerical values. By replacing variables with their corresponding values, we can calculate the outcome of the expression. This process is not only used in evaluating algebraic expressions but also in solving equations, graphing functions, and many other areas of mathematics. Accurate substitution is the cornerstone of algebraic manipulation, and mastering this step is essential for success in more advanced topics.
Step 2: Perform the Multiplication
Now that we have substituted the values, we need to perform the multiplication operations within the expression. According to the order of operations, multiplication should be done before addition and subtraction. Our expression currently looks like this:
((-5)(5) - 17(-5) - 5) / (-5)
Let's perform the multiplications:
- (-5)(5) = -25
- 17(-5) = -85
So, our expression now becomes:
(-25 - (-85) - 5) / (-5)
Multiplication is a critical step in simplifying expressions. It’s essential to remember the rules for multiplying signed numbers: a negative times a positive is negative, and a negative times a negative is positive. Attention to detail in this step ensures that the subsequent operations are performed on the correct values, leading to an accurate final result.
Multiplication: A Closer Look
Multiplication is one of the basic arithmetic operations and forms the basis for many algebraic manipulations. In this context, understanding how to multiply signed numbers is vital. The product of two negative numbers is positive, while the product of a positive and a negative number is negative. These rules are crucial in simplifying expressions and solving equations. Additionally, the distributive property, which involves multiplication over addition and subtraction, is a fundamental concept in algebra that relies on accurate multiplication.
Step 3: Simplify the Numerator
Next, we need to simplify the numerator of the expression. The numerator is the part of the fraction above the division line. Our expression currently looks like this:
(-25 - (-85) - 5) / (-5)
To simplify, we'll perform the subtraction and addition operations from left to right. First, let's handle the subtraction of a negative number:
-25 - (-85) = -25 + 85 = 60
Now, our expression inside the parentheses looks like this:
(60 - 5) / (-5)
Next, we subtract 5 from 60:
60 - 5 = 55
So, the simplified numerator is 55. Our expression now becomes:
55 / (-5)
Simplifying the numerator is a crucial step in reducing the expression to its simplest form. By correctly applying the rules of addition and subtraction, we ensure that the division step will yield the correct result. Paying attention to the order of operations and handling signed numbers accurately are key skills in this process.
The Importance of Order of Operations
The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), dictates the sequence in which operations should be performed in a mathematical expression. Following the correct order ensures that the expression is evaluated consistently and accurately. In this step, we simplified the numerator by first addressing the subtraction of a negative number and then performing the remaining subtraction. Adhering to the order of operations is fundamental for accurate mathematical calculations.
Step 4: Perform the Division
Finally, we perform the division operation to get our final simplified answer. Our expression has been simplified to:
55 / (-5)
Now, we divide 55 by -5:
55 / (-5) = -11
So, the final simplified answer is -11.
Performing the division is the last step in evaluating the expression. It’s essential to remember the rules for dividing signed numbers: a positive divided by a negative is negative, and a negative divided by a negative is positive. This step consolidates all the previous operations, leading to the ultimate result. Accuracy in this final step ensures that the entire evaluation process yields the correct answer.
Division and Its Role in Simplification
Division is one of the four basic arithmetic operations and is crucial in simplifying expressions and solving equations. In this context, understanding how to divide signed numbers is vital. The quotient of a positive number and a negative number is negative, while the quotient of two negative numbers is positive. These rules are fundamental in achieving the correct final result. Division also plays a key role in simplifying fractions and rational expressions, making it a cornerstone of algebraic manipulation.
Conclusion
We have successfully evaluated the expression (st - 17s - t) / s for s = -5 and t = 5, and the simplified answer is -11. By following the order of operations and carefully substituting the given values, we were able to break down the problem into manageable steps and arrive at the solution. Remember, practice makes perfect, so keep working on these types of problems to improve your algebra skills! If you're interested in learning more about algebraic expressions and evaluations, I highly recommend checking out resources from reputable websites such as Khan Academy Algebra, which offers comprehensive lessons and practice exercises.
