Six Times The Sum Of A Number And 15 Is

Six Times the Sum of a Number and 15 Is... A Deep Dive into Algebraic Expressions

This seemingly simple phrase, "six times the sum of a number and 15," opens up a world of mathematical exploration. It's a gateway to understanding algebraic expressions, equation solving, and even the foundations of more complex mathematical concepts. This article will dissect this phrase, exploring its algebraic representation, how to solve related equations, and its applications in various scenarios. We'll also touch upon the importance of understanding such expressions in real-world applications.

Understanding the Phrase: Breaking it Down

The phrase, "six times the sum of a number and 15," is a description of a mathematical operation. Let's break it down step by step:

• A number: This represents an unknown quantity, typically denoted by a variable like x, y, or n. For consistency, we'll use x throughout this article.

• The sum of a number and 15: This means adding 15 to our unknown number (x). Algebraically, this is expressed as x + 15.

• Six times the sum: This indicates multiplying the result of the previous step (x + 15) by 6. Algebraically, this becomes 6(x + 15).

Therefore, the complete algebraic expression for "six times the sum of a number and 15" is 6(x + 15). This is a crucial step; accurately translating word problems into algebraic expressions is fundamental to solving them.

Expanding the Expression: The Distributive Property

The expression 6(x + 15) can be simplified using the distributive property, a fundamental concept in algebra. The distributive property states that for any numbers a, b, and c: a(b + c) = ab + ac.

Applying this to our expression:

6(x + 15) = 6(x) + 6(15) = 6x + 90

This simplified expression, 6x + 90, is equivalent to the original phrase. Both expressions represent the same mathematical operation, but the second is often more convenient for solving equations.

Solving Equations: Finding the Value of 'x'

The phrase "six times the sum of a number and 15 is** something" sets up an equation. The "is" indicates equality. For example:

"Six times the sum of a number and 15 is 126" translates to the equation:

6(x + 15) = 126

To solve this equation, we can use the following steps:

1. Expand the expression: 6x + 90 = 126

2. Subtract 90 from both sides: 6x = 126 - 90 => 6x = 36

3. Divide both sides by 6: x = 36 / 6 => x = 6

Therefore, the number (x) is 6. We can verify this by substituting 6 back into the original equation:

6(6 + 15) = 6(21) = 126. This confirms our solution.

Different Scenarios and Equations

Let's explore different scenarios using variations of the phrase:

Scenario 1: Finding a Number when the Result is Unknown

"Six times the sum of a number and 15 is equal to a certain value, y." This translates to:

6(x + 15) = y

To solve for x, we need to know the value of y. The solution will be:

x = (y - 90) / 6

Scenario 2: Inequalities

Instead of equality, we can have inequalities. For example:

"Six times the sum of a number and 15 is greater than 100" translates to:

6(x + 15) > 100

Solving this inequality:

6x + 90 > 100 6x > 10 x > 10/6 x > 5/3

This means x must be greater than 5/3 (or approximately 1.67).

Scenario 3: Applications in Real-World Problems

This type of algebraic expression appears in various real-world scenarios. For instance:

• Profit Calculation: Imagine a business where the profit is six times the sum of the number of units sold (x) and a fixed cost of $15. The total profit (y) can be represented as: y = 6(x + 15). Knowing the profit, we can calculate the number of units sold.

• Geometry Problems: The area of a rectangle might be expressed as six times the sum of its length and a constant value (e.g., 15). We can use the expression to solve for unknown dimensions.

• Financial Modeling: In financial modeling, this type of expression could represent investments with compounding interest.

Expanding Mathematical Understanding: Beyond the Basics

Understanding the simple phrase "six times the sum of a number and 15" is more than just solving for x. It lays the groundwork for:

• Advanced Algebra: This forms the basis for understanding more complex algebraic expressions and equations, including quadratic equations, polynomial equations, and systems of equations.

• Calculus: The concepts of limits and derivatives, fundamental to calculus, build upon the algebraic foundations established by understanding expressions like this one.

• Problem-Solving Skills: Translating word problems into mathematical expressions is a crucial skill applicable in various fields, from science and engineering to business and finance.

Key Takeaways and Further Exploration

This in-depth analysis of a seemingly simple phrase highlights the power of algebraic expressions. We've explored the process of translating words into algebra, simplifying expressions, solving equations, and the real-world applications of such expressions. Remember these key points:

• Careful Translation: Precisely translate word problems into algebraic notation.

• Distributive Property: Master the distributive property for simplifying expressions.

• Equation Solving Techniques: Become proficient in solving various types of equations.

• Real-World Applications: Understand how algebraic expressions are used to model real-world scenarios.

For further exploration, consider studying different types of equations (linear, quadratic, etc.), exploring inequalities, and delving deeper into the real-world applications of algebra in your chosen field of study or interest. The journey from a simple phrase to a deep understanding of algebra is a rewarding one. The ability to translate real-world problems into mathematical expressions and solve them is a valuable skill that opens doors to many opportunities. Keep exploring, keep learning, and keep solving!