Find The Sum Of 8a+2b-4 And 3b-5. Write. In Expanded Form And Then Standard Form
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When working with algebraic expressions, it is essential to understand how to combine like terms effectively. This skill helps simplify complex expressions and solve equations efficiently. In this article, we will explore how to find the sum of two algebraic expressions: 8a + 2b - 4 and 3b - 5. We will break down the process step-by-step, first writing the sum in expanded form, then simplifying it into standard form. Whether you're a student learning algebra or someone brushing up on basic algebraic operations, this comprehensive guide will help you master these concepts.
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Understanding Algebraic Expressions
Before diving into the steps for adding the expressions, it's important to understand what algebraic expressions are.
What Are Algebraic Expressions?
- Algebraic expressions are mathematical phrases that contain variables (like a and b) and constants (numbers without variables).
- They can include addition, subtraction, multiplication, and division operations.
- Examples include 3a + 4b, 5x - 7, and 2y + 10.
Terms and Like Terms
- Terms are the parts of an algebraic expression separated by addition or subtraction signs.
- Like terms are terms that have the same variables raised to the same powers. For example:
- 8a and 3a are like terms.
- 2b and 3b are like terms.
- 8a and 2b are not like terms.
Step-by-Step Guide to Find the Sum
We are asked to find the sum of the two expressions:
- Expression 1: 8a + 2b - 4
- Expression 2: 3b - 5
Let's walk through the process systematically.
1. Write the Expressions in a Clear Format
Express the sum as:
\[
(8a + 2b - 4) + (3b - 5)
\]
2. Remove Parentheses and Combine Like Terms
Since the expressions are added, we can remove parentheses directly:
\[
8a + 2b - 4 + 3b - 5
\]
3. Group Like Terms
Group the terms based on their variables:
- Terms with 'a': \(8a\)
- Terms with 'b': \(2b + 3b\)
- Constants: \(-4 - 5\)
4. Simplify Each Group
- Combine the 'a' terms: \(8a\) (no other 'a' terms to combine)
- Combine the 'b' terms: \(2b + 3b = 5b\)
- Combine the constants: \(-4 - 5 = -9\)
5. Write the Final Expanded Expression
The sum in expanded form is:
\[
8a + 5b - 9
\]
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Converting to Standard Form
The standard form of a simplified algebraic expression arranges the terms in a specific order, usually with the highest degree (or variable) first.
What Is Standard Form?
- For linear expressions involving variables like a and b, the standard form typically orders the terms alphabetically: variable terms first, then constants.
- The standard form of the expression is:
Steps to Write in Standard Form
- Ensure like terms are combined, which we have already done.
- Arrange the terms in alphabetical order of the variables, i.e., 'a' before 'b'.
- Constants come at the end.
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Summary of the Process
| Step | Description | Result |
|---------|------------------------------------------------------|------------------------------|
| Write the expressions | Write the sum with parentheses | \((8a + 2b - 4) + (3b - 5)\) |
| Remove parentheses | Simplify the expression | \(8a + 2b - 4 + 3b - 5\) |
| Group like terms | Collect similar variables | \(8a + (2b + 3b) + (-4 - 5)\) |
| Simplify groups | Add coefficients and constants | \(8a + 5b - 9\) |
| Write in standard form | Arrange terms alphabetically | \(\boxed{8a + 5b - 9}\) |
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Additional Tips for Algebraic Addition
- Always identify and combine like terms — terms with the same variables raised to the same power.
- Pay attention to signs (+ or -) to avoid errors during addition.
- Constants are numbers without variables; combine them separately.
- When adding multiple expressions, write each clearly and proceed systematically.
Practical Applications of Combining Algebraic Expressions
Understanding how to add algebraic expressions is fundamental in various fields, including:
- Algebraic problem-solving: Simplifying expressions before solving equations.
- Calculus: Combining functions for derivatives or integrals.
- Physics: Summing forces or quantities expressed algebraically.
- Economics: Calculating total costs or revenues expressed in algebra.
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Practice Problems for Mastery
Try solving these to reinforce your understanding:
- Find the sum of \(5x + 3y - 2\) and \(4x - y + 7\). Write in expanded and standard form.
- Combine the expressions \(2m + 4n - 6\) and \(-m + 3n + 8\). Simplify completely.
- Given \(a^2 + 2ab - 3b^2\) and \(-a^2 + 4ab + b^2\), find their sum in standard form.
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Conclusion
Adding algebraic expressions involves a clear understanding of like terms, careful grouping, and proper arrangement. As demonstrated with the expressions \(8a + 2b - 4\) and \(3b - 5\), the process begins by expanding and grouping similar terms, then simplifying to arrive at the standard form: \(8a + 5b - 9\). Mastering these skills enhances your ability to handle more complex algebraic problems and builds a solid foundation for advanced mathematics.
Remember, practice is key. Regularly working through similar problems will help you become confident in manipulating algebraic expressions efficiently and accurately.
