Solve $\frac{1}{2} + \frac{1}{3}w = \frac{1}{6}$. Show Your Work.

Feb 27, 2025 by ADMIN 66 views

Introduction

In this article, we will be solving a linear equation with a variable. The equation is $\frac{1}{2} + \frac{1}{3}w = \frac{1}{6}$ . We will be using algebraic techniques to isolate the variable and find its value.

Step 1: Multiply Both Sides by the Least Common Multiple (LCM)

To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the LCM of 2, 3, and 6 is 6.

\frac{1}{2} + \frac{1}{3}w = \frac{1}{6}

Multiply both sides by 6:

6 \times \left(\frac{1}{2} + \frac{1}{3}w\right) = 6 \times \frac{1}{6}

This simplifies to:

3 + 2w = 1

Step 2: Isolate the Variable

Now that we have eliminated the fractions, we can isolate the variable by subtracting 3 from both sides of the equation.

3 + 2w - 3 = 1 - 3

This simplifies to:

2w = -2

Step 3: Solve for the Variable

To solve for the variable, we need to divide both sides of the equation by 2.

\frac{2w}{2} = \frac{-2}{2}

This simplifies to:

w = -1

Conclusion

In this article, we have solved a linear equation with a variable. We used algebraic techniques to isolate the variable and find its value. The final answer is $w = -1$ .

Example Use Case

This type of equation can be used in a variety of real-world applications, such as finance and engineering. For example, if we have a savings account with a balance of $100 and we deposit $50 per month, we can use this equation to find the number of months it will take to reach a balance of $200.

\frac{1}{2} + \frac{1}{3}w = \frac{1}{6}

In this case, the equation represents the balance of the account after a certain number of months. By solving for $w$ , we can find the number of months it will take to reach a balance of $200.

Tips and Tricks

When solving linear equations with variables, it's essential to follow the order of operations (PEMDAS) and to isolate the variable by adding, subtracting, multiplying, or dividing both sides of the equation by the same value.

Common Mistakes

One common mistake when solving linear equations with variables is to forget to multiply both sides of the equation by the LCM of the denominators. This can lead to incorrect solutions and incorrect conclusions.

Conclusion

Introduction

In our previous article, we solved a linear equation with a variable using algebraic techniques. In this article, we will answer some common questions related to solving linear equations with variables.

Q: What is a linear equation with a variable?

A: A linear equation with a variable is an equation that contains a variable (a letter or symbol that represents a value) and is in the form of ax + b = c, where a, b, and c are constants.

Q: How do I know if an equation is linear?

A: To determine if an equation is linear, look for the following characteristics:

The equation is in the form of ax + b = c.
The variable (x) is raised to the power of 1.
There are no squared or cubed terms.

Q: What is the least common multiple (LCM)?

A: The least common multiple (LCM) is the smallest multiple that two or more numbers have in common. In the context of solving linear equations with variables, the LCM is used to eliminate fractions by multiplying both sides of the equation by the LCM.

Q: How do I find the LCM of two or more numbers?

A: To find the LCM of two or more numbers, follow these steps:

List the multiples of each number.
Identify the smallest multiple that appears in both lists.
The LCM is the smallest multiple that appears in both lists.

Q: What is the order of operations (PEMDAS)?

A: The order of operations (PEMDAS) is a set of rules that dictate the order in which mathematical operations should be performed. The acronym PEMDAS stands for:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate any exponential expressions next.
Multiplication and Division: Evaluate multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate any addition and subtraction operations from left to right.

Q: How do I isolate the variable in a linear equation?

A: To isolate the variable in a linear equation, follow these steps:

Add or subtract the same value to both sides of the equation to eliminate any constants.
Multiply or divide both sides of the equation by the same value to eliminate any fractions.
Continue to simplify the equation until the variable is isolated.

Q: What are some common mistakes to avoid when solving linear equations with variables?

A: Some common mistakes to avoid when solving linear equations with variables include:

Forgetting to multiply both sides of the equation by the LCM.
Not following the order of operations (PEMDAS).
Not isolating the variable correctly.

Conclusion

In conclusion, solving linear equations with variables is a fundamental concept in algebra. By following the steps outlined in this article and avoiding common mistakes, you can solve equations of this type and find the value of the variable.

Example Use Case

\frac{1}{2} + \frac{1}{3}w = \frac{1}{6}

In this case, the equation represents the balance of the account after a certain number of months. By solving for $w$ , we can find the number of months it will take to reach a balance of $200.