Translate And Simplify The Sum Of 7 And -19 Increased By 13

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This article will guide you through translating a verbal statement into an algebraic expression and simplifying it completely. We'll focus on the specific example: "The sum of 7 and -19, increased by 13." This is a fundamental skill in algebra, bridging the gap between everyday language and mathematical notation. Mastering this skill is crucial for solving more complex problems and understanding mathematical concepts.

Breaking Down the Statement

To translate the statement effectively, we need to break it down into smaller, manageable parts. Let's analyze each phrase:

  • "The sum of 7 and -19": This indicates an addition operation between the numbers 7 and -19. In mathematical notation, this is represented as 7 + (-19).
  • "Increased by 13": This phrase signifies that we need to add 13 to the result of the previous operation. So, we'll be adding 13 to the sum of 7 and -19.

Now that we've dissected the statement, we can begin constructing the algebraic expression.

Constructing the Algebraic Expression

Based on our breakdown, we can now write the algebraic expression. We know that "the sum of 7 and -19" is represented as 7 + (-19). Then, we need to increase this sum by 13, which means adding 13 to the expression. Therefore, the complete algebraic expression is:

(7 + (-19)) + 13

Notice the use of parentheses. In this case, they are used for clarity to group the first operation (the sum of 7 and -19). While the order of operations (PEMDAS/BODMAS) would dictate that addition is performed from left to right anyway, the parentheses explicitly show the intended grouping. This practice is especially useful when dealing with more complex expressions involving different operations.

Simplifying the Expression

Our next step is to simplify the algebraic expression we've constructed. This involves performing the operations in the correct order. According to the order of operations (PEMDAS/BODMAS), we handle parentheses first. In this case, we have the expression within the parentheses: 7 + (-19).

Step 1: Adding 7 and -19

Adding a positive number and a negative number can be thought of as finding the difference between their absolute values and taking the sign of the number with the larger absolute value. The absolute value of 7 is |7| = 7, and the absolute value of -19 is |-19| = 19. The difference between 19 and 7 is 12. Since -19 has a larger absolute value, the result of 7 + (-19) is -12.

So, our expression now becomes:

-12 + 13

Step 2: Adding -12 and 13

Now we have another addition of a negative and a positive number. Again, we find the difference between their absolute values: |13| = 13 and |-12| = 12. The difference is 13 - 12 = 1. Since 13 has a larger absolute value and is positive, the result is positive 1.

Therefore, the simplified expression is:

1

The Final Answer

After translating the statement "The sum of 7 and -19, increased by 13" into an algebraic expression and simplifying it completely, we arrive at the final answer:

1

This demonstrates the process of converting verbal statements into mathematical language and then using the rules of algebra to find a solution. This skill is fundamental for success in algebra and other areas of mathematics.

Key Concepts Revisited

Let's reinforce the key concepts we've covered in this example:

  • Translating verbal statements: This involves carefully dissecting the statement to identify the mathematical operations and the numbers involved. Key phrases like "sum," "increased by," "difference," and "product" indicate specific operations.
  • Algebraic expressions: These are combinations of numbers, variables, and mathematical operations. They provide a concise way to represent mathematical relationships.
  • Order of operations (PEMDAS/BODMAS): This set of rules dictates the order in which operations are performed in an expression. Parentheses/Brackets are evaluated first, then Exponents/Orders, then Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right).
  • Simplifying expressions: This involves performing the operations in the correct order to reduce the expression to its simplest form.

Understanding these concepts is crucial for building a strong foundation in algebra.

Practice Problems

To further solidify your understanding, try translating and simplifying the following statements:

  1. The difference between 15 and -8, decreased by 4.
  2. The product of -3 and 6, increased by 10.
  3. The sum of -20 and 5, divided by -3.

Working through these practice problems will help you develop your skills in translating verbal statements and simplifying algebraic expressions. Remember to break down each statement into smaller parts, identify the operations involved, and follow the order of operations.

Conclusion

Translating verbal statements into algebraic expressions and simplifying them is a fundamental skill in algebra. By carefully breaking down the statements, identifying the operations, and following the order of operations, you can successfully solve these types of problems. The example we worked through, "The sum of 7 and -19, increased by 13," demonstrates the process step-by-step. With practice and a solid understanding of the key concepts, you can confidently tackle more complex algebraic problems.

This skill of translating words into mathematical expressions is not just limited to the classroom. It's a crucial skill for problem-solving in various real-world scenarios, from budgeting and finance to engineering and science. The ability to think quantitatively and translate real-world problems into mathematical models allows for effective analysis and decision-making.

Continue practicing and exploring different types of problems to further enhance your algebraic skills. The journey through mathematics is a continuous process of learning and building upon foundational concepts. The more you practice, the more confident and proficient you will become.