Six Times The Sum Of A Number And 2.

Six Times the Sum of a Number and 2: A Deep Dive into Algebraic Expressions

This article delves into the algebraic expression "six times the sum of a number and 2," exploring its various interpretations, applications, and how to solve problems related to it. We'll move beyond simple calculations and uncover the underlying mathematical principles, demonstrating its relevance in various real-world scenarios. We will also cover advanced topics such as solving equations involving this expression and exploring its graphical representation.

Understanding the Expression

The phrase "six times the sum of a number and 2" can be translated directly into an algebraic expression. Let's break it down step-by-step:

• A number: We'll represent this unknown number with the variable x.
• The sum of a number and 2: This translates to x + 2.
• Six times the sum of a number and 2: This becomes 6(x + 2). Notice the use of parentheses; they are crucial to indicate that the entire sum (x + 2) is multiplied by 6.

Therefore, the algebraic representation of the phrase is 6(x + 2). This seemingly simple expression opens the door to a world of mathematical exploration.

Expanding the Expression

Often, it's beneficial to expand (or simplify) an algebraic expression. In this case, we can use the distributive property of multiplication to remove the parentheses:

6(x + 2) = 6 * x + 6 * 2 = 6x + 12

This expanded form, 6x + 12, is equivalent to the original expression 6(x + 2). Both expressions represent the same mathematical concept, but the expanded form is often easier to work with in certain calculations and problem-solving scenarios.

Solving Equations Involving the Expression

The real power of this expression comes into play when it's part of an equation. Let's consider a few examples:

Example 1: Finding the Unknown Number

Let's say the expression 6(x + 2) is equal to 30. We can write this as an equation:

6(x + 2) = 30

To solve for x, we follow these steps:

1. Expand the expression: 6x + 12 = 30
2. Subtract 12 from both sides: 6x = 18
3. Divide both sides by 6: x = 3

Therefore, the unknown number (x) is 3. We can check our answer by substituting x = 3 back into the original equation: 6(3 + 2) = 6(5) = 30. This confirms our solution.

Example 2: A More Complex Equation

Consider a slightly more complex equation:

2[6(x + 2)] + 4 = 40

Here's how to solve this step-by-step:

1. Expand the inner expression: 2[6x + 12] + 4 = 40
2. Expand the outer expression: 12x + 24 + 4 = 40
3. Combine like terms: 12x + 28 = 40
4. Subtract 28 from both sides: 12x = 12
5. Divide both sides by 12: x = 1

Again, we can check our work by substituting x = 1 into the original equation. This will verify that our solution is correct.

Example 3: Equations with Variables on Both Sides

Let's look at an equation with the expression on both sides:

6(x + 2) = 3x + 18

Solving this requires slightly different steps:

1. Expand the expression: 6x + 12 = 3x + 18
2. Subtract 3x from both sides: 3x + 12 = 18
3. Subtract 12 from both sides: 3x = 6
4. Divide both sides by 3: x = 2

Example 4: Word Problems

The true test of understanding lies in applying the algebraic expression to word problems. For instance:

"John earns six times the sum of his hourly rate and $2 bonus per hour. If he earns $36 per hour, what is his hourly rate?"

Let r represent John's hourly rate. We can translate this into the equation:

6(r + 2) = 36

Solving this equation using the methods outlined above will reveal John's hourly rate.

Graphical Representation

The expression 6(x + 2), or its simplified form 6x + 12, can be graphically represented as a straight line. This is because it's a linear equation. The graph will have a slope of 6 and a y-intercept of 12. Understanding this graphical representation provides a visual interpretation of the relationship between x and the value of the expression. The slope indicates the rate of change, showing how the expression's value increases with each unit increase in x. The y-intercept represents the value of the expression when x is 0.

Plotting points on a coordinate plane, such as (0,12), (1,18), (2,24), etc., and connecting them will reveal the straight line representing the expression. This visual representation aids in understanding the behavior of the expression and helps in solving related problems.

Applications in Real-World Scenarios

The expression "six times the sum of a number and 2" might seem abstract, but it has numerous real-world applications:

• Calculating total earnings: As demonstrated in the word problem above, it can model scenarios involving bonuses, commissions, or overtime pay.
• Determining total cost: It can represent the total cost of items where there's a base price plus a per-item cost, multiplied by a certain quantity.
• Analyzing growth patterns: In certain contexts, it could represent exponential growth or linear growth scenarios where the initial value is increased by a constant factor.
• Simple interest calculations: Although not a perfect analogy, it can be a simplified way to think about simple interest earned over multiple periods.

By understanding this seemingly simple expression, we can model and solve a range of real-world problems.

Advanced Topics: Inequalities and More Complex Equations

This expression can also be used in inequalities. For instance, "six times the sum of a number and 2 is greater than 30" can be written as:

6(x + 2) > 30

Solving this inequality involves similar steps to solving equations, but the solution will be a range of values for x rather than a single value.

Furthermore, the expression can be incorporated into much more complex equations involving multiple variables and operations. The fundamental principles of solving equations – expanding, simplifying, isolating the variable – remain the same, regardless of complexity.

Conclusion

The seemingly simple algebraic expression "six times the sum of a number and 2" unlocks a wide range of mathematical possibilities. From solving basic equations to tackling complex word problems and visualizing the expression graphically, its applications are diverse and relevant in various fields. A thorough understanding of this expression provides a strong foundation for more advanced algebraic concepts and problem-solving techniques. By mastering this fundamental concept, you equip yourself with valuable tools for tackling more challenging mathematical challenges in the future. Remember the key steps: translate the phrase into an algebraic expression, expand and simplify using the distributive property, and then solve for the unknown using appropriate algebraic techniques. Practice makes perfect, so keep working through examples and word problems to strengthen your understanding.