Equation For Six Added To Twice The Sum Of A Number And Four

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In the realm of mathematics, translating word problems into algebraic equations is a fundamental skill. This article delves into the process of converting the statement "six added to twice the sum of a number and four is equal to one-half of the difference of three and the number" into its corresponding equation. We will dissect the statement, break it down into smaller parts, and then reconstruct it in mathematical notation. This exercise not only reinforces the understanding of algebraic principles but also highlights the importance of precise language in mathematical communication.

Deconstructing the Statement: A Step-by-Step Approach

To effectively translate the given statement into an equation, we need to carefully analyze each phrase and identify the mathematical operations and relationships they represent. Let's break down the statement into smaller, manageable parts:

  1. "a number": This phrase represents an unknown quantity, which we can denote using a variable. Let's use the variable x to represent this unknown number.
  2. "the sum of a number and four": This phrase indicates addition. We are adding the number x to four, which can be written as (x + 4).
  3. "twice the sum of a number and four": This phrase implies multiplication. We are multiplying the sum (x + 4) by two, which can be written as 2(x + 4).
  4. "six added to twice the sum of a number and four": This phrase indicates addition again. We are adding six to the expression 2(x + 4), which can be written as 2(x + 4) + 6.
  5. "the difference of three and the number": This phrase indicates subtraction. We are subtracting the number x from three, which can be written as (3 - x). Note that the order of subtraction matters.
  6. "one-half of the difference of three and the number": This phrase implies multiplication by a fraction. We are multiplying the difference (3 - x) by one-half, which can be written as (1/2)(3 - x) or (3 - x)/2.
  7. "is equal to": This phrase represents the equals sign (=), which connects the two sides of the equation.

Reconstructing the Equation: Putting the Pieces Together

Now that we have analyzed each part of the statement, we can reconstruct it into a complete equation. Combining the expressions we derived in the previous section, we get:

2(x + 4) + 6 = (3 - x)/2

This equation accurately represents the original statement. It states that six added to twice the sum of a number and four is equal to one-half of the difference of three and the number.

Exploring Alternative Representations

While the equation 2(x + 4) + 6 = (3 - x)/2 is a valid representation, it can be written in different forms that are mathematically equivalent. For example, we can distribute the 2 on the left side and multiply both sides of the equation by 2 to eliminate the fraction. Let's explore these alternative representations:

Distributing the 2:

We can distribute the 2 in the expression 2(x + 4) by multiplying it with both terms inside the parentheses:

2(x + 4) = 2x + 24 = 2x + 8

Substituting this back into the original equation, we get:

2x + 8 + 6 = (3 - x)/2

Simplifying the left side, we have:

2x + 14 = (3 - x)/2

Eliminating the Fraction:

To eliminate the fraction, we can multiply both sides of the equation by 2:

2(2x + 14) = 2[(3 - x)/2]

This simplifies to:

4x + 28 = 3 - x

This equation is equivalent to the original equation but does not contain any fractions.

Solving the Equation: Finding the Value of x

Now that we have the equation in different forms, we can solve for the unknown variable x. Let's solve the equation 4x + 28 = 3 - x:

  1. Add x to both sides: This will eliminate the x term on the right side: 4x + x + 28 = 3 - x + x 5x + 28 = 3
  2. Subtract 28 from both sides: This will isolate the x term on the left side: 5x + 28 - 28 = 3 - 28 5x = -25
  3. Divide both sides by 5: This will solve for x: 5x/5 = -25/5 x = -5

Therefore, the solution to the equation is x = -5. This means that if we substitute -5 for x in the original equation, both sides will be equal.

Verifying the Solution: Ensuring Accuracy

To ensure that our solution is correct, we can substitute x = -5 back into the original equation and check if both sides are equal:

Original equation: 2(x + 4) + 6 = (3 - x)/2

Substitute x = -5:

2(-5 + 4) + 6 = (3 - (-5))/2

Simplify:

2(-1) + 6 = (3 + 5)/2

-2 + 6 = 8/2

4 = 4

Since both sides of the equation are equal, our solution x = -5 is correct.

Importance of Precision in Mathematical Language

This exercise highlights the importance of precision in mathematical language. Each word and phrase carries a specific mathematical meaning, and any ambiguity can lead to an incorrect equation. By carefully analyzing the statement and breaking it down into smaller parts, we can ensure that we accurately capture the intended mathematical relationships. This skill is crucial for solving word problems and communicating mathematical ideas effectively.

Real-World Applications

Translating word problems into algebraic equations is a fundamental skill that has numerous real-world applications. It is used in various fields, including:

  • Finance: Calculating interest rates, loan payments, and investment returns.
  • Engineering: Designing structures, analyzing circuits, and modeling physical systems.
  • Physics: Solving problems related to motion, forces, and energy.
  • Computer Science: Developing algorithms, creating simulations, and analyzing data.

By mastering the skill of translating word problems into equations, individuals can effectively solve problems in these and other fields.

Conclusion

Translating the statement "six added to twice the sum of a number and four is equal to one-half of the difference of three and the number" into an equation involves careful analysis of the language and understanding of mathematical operations. The equation 2(x + 4) + 6 = (3 - x)/2 accurately represents the given statement. By breaking down the statement into smaller parts, we can identify the mathematical operations and relationships they represent and then reconstruct them into a complete equation. This skill is essential for solving word problems and communicating mathematical ideas effectively. The solution to the equation is x = -5, which can be verified by substituting it back into the original equation. This exercise underscores the significance of precision in mathematical language and its real-world applications in various fields.

By understanding the process of translating word problems into equations, we can unlock the power of algebra to solve a wide range of problems and gain a deeper appreciation for the beauty and precision of mathematics. This skill is not only valuable in academic settings but also in everyday life, where we often encounter situations that require us to think critically and solve problems using mathematical reasoning.

Practice Problems

To further solidify your understanding of translating word problems into equations, try solving the following practice problems:

  1. Five less than three times a number is equal to the sum of the number and seven. Write an equation to represent this statement and solve for the number.
  2. The quotient of a number and two, increased by four, is equal to twice the number. Write an equation to represent this statement and solve for the number.
  3. The sum of two consecutive integers is 35. Write an equation to represent this statement and find the two integers.

By working through these practice problems, you can strengthen your skills in translating word problems into equations and become more confident in your ability to solve algebraic problems.