Solve: $\frac{3}{2} X + \frac{11}{6} = \frac{1}{3}$x =$ $\square$

Feb 28, 2025 by ADMIN 67 views

Solving Linear Equations: A Step-by-Step Guide to Isolating Variables

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students and professionals alike. In this article, we will focus on solving a specific type of linear equation, which involves isolating the variable on one side of the equation. We will use the given equation $\frac{3}{2} x + \frac{11}{6} = \frac{1}{3} x$ as an example to demonstrate the step-by-step process.

Understanding the Equation

Before we dive into solving the equation, let's take a closer look at its components. The equation is in the form of a linear equation, which is a polynomial equation of degree one. The equation has two fractions on the left-hand side and one fraction on the right-hand side. Our goal is to isolate the variable $x$ on one side of the equation.

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, 6, and 3 is 6. By multiplying both sides by 6, we can eliminate the fractions.

\frac{3}{2} x + \frac{11}{6} = \frac{1}{3} x
\implies 6 \left( \frac{3}{2} x + \frac{11}{6} \right) = 6 \left( \frac{1}{3} x \right)
\implies 9x + 11 = 2x

Step 2: Subtract 2x from Both Sides

Now that we have eliminated the fractions, we can focus on isolating the variable $x$ . To do this, we need to subtract $2x$ from both sides of the equation. This will help us get all the terms with $x$ on one side of the equation.

9x + 11 = 2x
\implies 9x - 2x + 11 = 2x - 2x
\implies 7x + 11 = 0

Step 3: Subtract 11 from Both Sides

Next, we need to subtract 11 from both sides of the equation. This will help us get all the constant terms on one side of the equation.

7x + 11 = 0
\implies 7x + 11 - 11 = 0 - 11
\implies 7x = -11

Step 4: Divide Both Sides by 7

Finally, we need to divide both sides of the equation by 7. This will help us isolate the variable $x$ and solve for its value.

7x = -11
\implies \frac{7x}{7} = \frac{-11}{7}
\implies x = -\frac{11}{7}

Conclusion

In this article, we have demonstrated the step-by-step process of solving a linear equation. We started by multiplying both sides of the equation by the least common multiple (LCM) of the denominators, then subtracted $2x$ from both sides, subtracted 11 from both sides, and finally divided both sides by 7. By following these steps, we were able to isolate the variable $x$ and solve for its value.

Tips and Tricks

When solving linear equations, it's essential to follow the order of operations (PEMDAS) to ensure that you are performing the operations in the correct order.
When multiplying both sides of the equation by the LCM, make sure to multiply both sides by the same value.
When subtracting or adding terms to both sides of the equation, make sure to perform the operation on both sides of the equation.
When dividing both sides of the equation by a value, make sure to divide both sides by the same value.

Frequently Asked Questions

Q: What is the least common multiple (LCM) of 2, 6, and 3? A: The LCM of 2, 6, and 3 is 6.
Q: How do I multiply both sides of the equation by the LCM? A: To multiply both sides of the equation by the LCM, simply multiply both sides by the LCM value.
Q: How do I subtract $2x$ from both sides of the equation? A: To subtract $2x$ from both sides of the equation, simply subtract $2x$ from both sides of the equation.
Q: How do I divide both sides of the equation by 7? A: To divide both sides of the equation by 7, simply divide both sides of the equation by 7.

Final Thoughts

Solving linear equations is a crucial skill for students and professionals alike. By following the step-by-step process outlined in this article, you can isolate the variable $x$ and solve for its value. Remember to follow the order of operations (PEMDAS), multiply both sides of the equation by the LCM, subtract $2x$ from both sides, subtract 11 from both sides, and finally divide both sides by 7. With practice and patience, you can become proficient in solving linear equations and tackle even the most challenging problems.

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Introduction

In our previous article, we demonstrated the step-by-step process of solving a linear equation. However, we understand that sometimes, it's not just about following a set of steps, but also about understanding the underlying concepts and addressing common questions and concerns. In this article, we will provide a Q&A guide to help you better understand linear equations and how to solve them.

Q: What is a linear equation?

A: A linear equation is a polynomial equation of degree one, which means that the highest power of the variable is one. Linear equations are typically written in the form of ax + b = c, where a, b, and c are constants, and x is the variable.

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

A: The least common multiple (LCM) of a set of numbers is the smallest number that is a multiple of each of the numbers in the set. In the context of linear equations, 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 a set of numbers?

A: To find the LCM of a set of numbers, you can use the following steps:

List the multiples of each number in the set.
Identify the smallest number that is a multiple of each number in the set.
The LCM is the smallest number that is a multiple of each number in the set.

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 any 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 multiply both sides of the equation by the LCM?

A: To multiply both sides of the equation by the LCM, simply multiply both sides of the equation by the LCM value. For example, if the LCM is 6, you would multiply both sides of the equation by 6.

Q: How do I subtract 2x from both sides of the equation?

A: To subtract 2x from both sides of the equation, simply subtract 2x from both sides of the equation. For example, if the equation is 9x + 11 = 2x, you would subtract 2x from both sides to get 7x + 11 = 0.

Q: How do I divide both sides of the equation by 7?

A: To divide both sides of the equation by 7, simply divide both sides of the equation by 7. For example, if the equation is 7x = -11, you would divide both sides by 7 to get x = -11/7.

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

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

Not following the order of operations (PEMDAS)
Not multiplying both sides of the equation by the LCM
Not subtracting 2x from both sides of the equation
Not dividing both sides of the equation by the correct value

Q: How can I practice solving linear equations?

A: There are many ways to practice solving linear equations, including:

Using online resources and practice problems
Working with a tutor or teacher
Practicing with real-world examples and applications
Joining a study group or math club

Conclusion

Solving linear equations is a crucial skill for students and professionals alike. By understanding the underlying concepts and addressing common questions and concerns, you can become proficient in solving linear equations and tackle even the most challenging problems. Remember to follow the order of operations (PEMDAS), multiply both sides of the equation by the LCM, subtract 2x from both sides, subtract 11 from both sides, and finally divide both sides by 7. With practice and patience, you can become a master of solving linear equations.

Final Thoughts

Solving linear equations is not just about following a set of steps, but also about understanding the underlying concepts and addressing common questions and concerns. By practicing regularly and seeking help when needed, you can become proficient in solving linear equations and tackle even the most challenging problems. Remember to stay focused, persistent, and patient, and you will be solving linear equations like a pro in no time!