Three Less Than Six Times A Number

Three Less Than Six Times a Number: A Deep Dive into Mathematical Expressions

This article delves into the mathematical expression "three less than six times a number," exploring its various interpretations, applications, and significance in algebra and beyond. We'll unpack its meaning, demonstrate how to translate it into algebraic notation, solve related equations, and even explore real-world applications. This comprehensive guide is designed to enhance your understanding of mathematical expressions and their practical uses.

Understanding the Expression: "Three Less Than Six Times a Number"

The phrase "three less than six times a number" represents a specific mathematical operation. Let's break it down step-by-step:

• "A number": This represents an unknown value, commonly denoted by a variable like x, y, or n.

• "Six times a number": This translates to multiplying the unknown number by six, which is algebraically represented as 6x (or 6n, 6y, etc.).

• "Three less than": This indicates subtracting three from the previous result (6x).

Therefore, the complete expression "three less than six times a number" is mathematically written as 6x - 3. This is a simple algebraic expression, yet it forms the basis for many more complex equations and problems.

Translating Words into Algebraic Equations

The ability to translate word problems into algebraic equations is a fundamental skill in mathematics. This process involves carefully identifying the key phrases and translating them into mathematical symbols and operations. Understanding the order of operations (PEMDAS/BODMAS) is crucial here. In our case, "six times a number" takes precedence over "three less than," leading to the subtraction being performed last.

Let's look at a few examples to solidify this understanding:

Example 1: "Five more than twice a number" translates to 2x + 5.

Example 2: "The product of a number and seven, decreased by four" translates to 7x - 4.

Example 3: "Ten less than the square of a number" translates to x² - 10.

These examples highlight the importance of paying close attention to the wording of the problem. The order of operations and the precise meaning of phrases like "more than," "less than," and "decreased by" are critical to forming the correct equation.

Solving Equations Involving "Three Less Than Six Times a Number"

Once we have translated the word problem into an algebraic equation, we can solve for the unknown variable. This often involves using inverse operations to isolate the variable. Let's consider an example:

Problem: "Three less than six times a number is 15. Find the number."

Solution:

1. Translate the problem into an equation: 6x - 3 = 15

2. Add 3 to both sides of the equation: 6x = 18

3. Divide both sides by 6: x = 3

Therefore, the number is 3. We can check our answer by substituting it back into the original equation: 6(3) - 3 = 15, which is true.

Let's try another example with a slightly different approach:

Problem: "If three less than six times a number is equal to twice the number plus seven, find the number."

Solution:

1. Translate the problem into an equation: 6x - 3 = 2x + 7

2. Subtract 2x from both sides: 4x - 3 = 7

3. Add 3 to both sides: 4x = 10

4. Divide both sides by 4: x = 2.5

In this case, the solution is a decimal, demonstrating that algebraic equations can produce solutions that are not always whole numbers.

Real-World Applications of the Expression

While seemingly simple, the expression "three less than six times a number" (and its related equations) has several real-world applications in various fields:

1. Geometry: Calculating areas or perimeters of shapes often involves using algebraic expressions. For example, finding the length of a side given the area and other dimensions of a rectangle might lead to an equation resembling our core expression.

2. Physics: Many physics problems involve translating word descriptions into mathematical relationships. Determining velocity, acceleration, or distance given specific conditions can often lead to equations that include similar algebraic components.

3. Finance: Calculating simple interest, discounts, or profits often involves operations that mirror the structure of our expression. For example, finding the final cost after a discount could be formulated as an equation involving multiplication (like "six times a number") and subtraction (like "three less than").

4. Business: Businesses frequently employ algebraic expressions in cost analysis, profit margin calculations, or sales forecasting.

5. Computer Programming: Writing algorithms and performing calculations within computer programs inevitably requires using algebraic expressions and variables, the bedrock of which are simple expressions like the one we are studying.

Expanding the Concept: More Complex Scenarios

The core concept of "three less than six times a number" can be extended to more complex scenarios. For instance:

• Inequalities: Instead of an equation (using an equals sign), we could have an inequality (using >, <, ≥, ≤). For example: "Three less than six times a number is greater than 15." This would translate to 6x - 3 > 15, requiring a different solution approach.

• Systems of Equations: We could have multiple equations involving this expression along with others. This would require solving a system of equations to find the solution.

• Functions: This expression can be incorporated into a function, allowing for more dynamic calculations and analysis. For instance, f(x) = 6x - 3 defines a linear function.

• Graphing: This function can be graphed, allowing for a visual representation of the relationship between the input (x) and output (y=6x -3) values. The graph will be a straight line, reflecting the linear nature of the function.

Conclusion: Mastering Mathematical Expressions for Real-World Success

The seemingly simple expression "three less than six times a number" serves as a powerful entry point into the world of algebra and its wide-ranging applications. By understanding how to translate words into equations, solve for unknowns, and apply these concepts to real-world problems, you build a strong foundation for success in mathematics and related fields. This comprehensive understanding extends far beyond simple equations, impacting your ability to tackle complex mathematical challenges, solve problems effectively, and interpret information critically. The skill of translating word problems into mathematical equations is a fundamental transferable skill valued across numerous disciplines. Remember to practice regularly, explore different problem types, and always break down complex problems into smaller, manageable steps.