What Is The Sum Of These Expressions: (-3.5t - 4s + 4.5) + (-7.1 - 0.3s + 4.1t)

What Is The Sum Of These Expressions: (-3.5t - 4s + 4.5) + (-7.1 - 0.3s + 4.1t)

Understanding how to simplify algebraic expressions is fundamental in mathematics, especially when dealing with multiple variables. The question "What is the sum of these expressions: (-3.5t - 4s + 4.5) + (-7.1 - 0.3s + 4.1t)" presents an excellent opportunity to explore the process of combining like terms, handling coefficients, and simplifying algebraic expressions for clarity and further use. This article will walk you through the detailed steps involved in solving this problem, explain key concepts in algebra, and provide tips for tackling similar problems efficiently.

Breaking Down the Problem: Understanding the Expressions

Before diving into the calculations, it is essential to understand the structure of the given expressions.

The Expressions to Add

The two algebraic expressions are:

1. First Expression: -3.5t - 4s + 4.5
2. Second Expression: -7.1 - 0.3s + 4.1t

The goal is to find their sum, which involves combining like terms—terms that contain the same variables raised to the same powers.

Variables and Constants

In these expressions, we encounter:

• Variables: t and s
• Constants: numbers without variables, such as 4.5 and -7.1

When adding these expressions, only like terms—those with the same variables—are combined. Constants are combined separately.

Step-by-Step Solution Approach

To find the sum of the two expressions, follow these steps:

1. Write the expressions clearly

Express the sum as:
\[
(-3.5t - 4s + 4.5) + (-7.1 - 0.3s + 4.1t)
\]

1. Remove parentheses

Since both expressions are added, the parentheses can be removed, but keep the signs in mind:
\[
-3.5t - 4s + 4.5 - 7.1 - 0.3s + 4.1t
\]

1. Group like terms

Group all terms containing the same variables and constants:

• Terms with t: \(-3.5t\) and \(4.1t\)


• Terms with s: \(-4s\) and \(-0.3s\)


• Constants: \(4.5\) and \(-7.1\)

1. Combine like terms

Adding the coefficients of similar terms:

For t:
\[
-3.5t + 4.1t = ( -3.5 + 4.1 ) t = 0.6t
\]

For s:
\[
-4s - 0.3s = ( -4 - 0.3 ) s = -4.3s
\]

Constants:
\[
4.5 - 7.1 = -2.6
\]

1. Write the simplified sum

Putting these together:
\[
0.6t - 4.3s - 2.6
\]

This is the simplified form of the sum of the two expressions.

Understanding the Importance of Combining Like Terms

Combining like terms is a crucial skill in algebra because it simplifies expressions, making them easier to evaluate, graph, or substitute values into. It helps in solving equations, analyzing functions, and performing algebraic manipulations efficiently.

Why Are Like Terms Important?

• They ensure clarity and simplicity in expressions.
• They allow for straightforward addition, subtraction, and further algebraic operations.
• They help in solving equations, as it reduces the complexity of the problem.

Additional Tips for Simplifying Algebraic Expressions

When dealing with algebraic expressions, keep these tips in mind:

• Always identify like terms: Variables with the same base and exponents are like terms.
• Pay attention to signs: Remember that adding a negative is equivalent to subtraction.
• Line up similar terms: Organize your expression to make combining terms easier.
• Check your work: After combining, verify that you have included all terms and kept track of signs.

Applying This Knowledge to Real-World Problems

Algebraic expressions are foundational in various fields such as physics, engineering, economics, and computer science. Simplifying expressions allows professionals to model real-world scenarios, optimize solutions, and interpret data effectively.

Example: If t represents time in hours and s represents speed in miles per hour, the simplified expression \(0.6t - 4.3s - 2.6\) could be used to model a particular relationship or scenario, like estimating fuel consumption or travel time adjustments.

Practice Exercises to Master Combining Like Terms

To reinforce understanding, consider practicing with similar problems:

• Find the sum of: (2x + 3y - 4) + (-x + 5y + 6)
• Simplify: (5a - 2b + 7) + (-3a + 4b - 2)
• Combine: (7m + 2n - 3) + (-4m + 6n + 8)

By working through these and similar exercises, you'll develop confidence and proficiency in algebraic manipulation.

Conclusion

The process of finding the sum of the expressions \((-3.5t - 4s + 4.5) + (-7.1 - 0.3s + 4.1t)\) exemplifies core algebra skills: identifying like terms, combining coefficients, and simplifying expressions for clarity. The final simplified form, \(0.6t - 4.3s - 2.6\), is easier to interpret, evaluate, or substitute specific values into for further calculations.

Mastering these techniques enhances problem-solving skills and provides a solid foundation for more advanced mathematical topics. Whether you're solving equations, modeling real-world situations, or preparing for exams, understanding how to simplify algebraic expressions is an invaluable skill in mathematics and beyond.

Remember: Practice makes perfect. Keep working on similar problems, and you'll become more confident in handling algebraic expressions with ease.