Select The Correct Answer.What Is The Solution For $x$ In The Equation?$\frac{5}{3} X + 4 = \frac{2}{3} X$A. $x = -\frac{12}{7}$ B. $x = \frac{12}{7}$ C. $x = -4$ D. $x = 4$

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Introduction

Linear equations are a fundamental concept in mathematics, and solving them is a crucial skill for students to master. In this article, we will focus on solving a specific type of linear equation, which involves isolating the variable x. We will use the equation 53x+4=23x\frac{5}{3} x + 4 = \frac{2}{3} x as an example and guide you through the step-by-step process of solving for x.

Understanding the Equation

The given equation is 53x+4=23x\frac{5}{3} x + 4 = \frac{2}{3} x. To solve for x, we need to isolate the variable x on one side of the equation. The first step is to simplify the equation by getting rid of the fractions.

Step 1: Eliminate the Fractions

To eliminate the fractions, we can multiply both sides of the equation by the least common multiple (LCM) of the denominators, which is 3.

53x+4=23x\frac{5}{3} x + 4 = \frac{2}{3} x

Multiplying both sides by 3:

5x+12=2x5x + 12 = 2x

Step 2: Isolate the Variable x

Now that we have eliminated the fractions, we can isolate the variable x by subtracting 2x from both sides of the equation.

5x+12=2x5x + 12 = 2x

Subtracting 2x from both sides:

3x+12=03x + 12 = 0

Step 3: Solve for x

To solve for x, we need to isolate x on one side of the equation. We can do this by subtracting 12 from both sides of the equation.

3x+12=03x + 12 = 0

Subtracting 12 from both sides:

3x=−123x = -12

Step 4: Divide by the Coefficient

Finally, we can solve for x by dividing both sides of the equation by the coefficient of x, which is 3.

3x=−123x = -12

Dividing both sides by 3:

x=−123x = -\frac{12}{3}

Simplifying the fraction:

x=−4x = -4

Conclusion

In this article, we have solved the linear equation 53x+4=23x\frac{5}{3} x + 4 = \frac{2}{3} x by following a step-by-step process. We eliminated the fractions, isolated the variable x, and solved for x. The solution to the equation is x=−4x = -4. This example demonstrates the importance of following a systematic approach when solving linear equations.

Answer

The correct answer is:

  • A. x=−127x = -\frac{12}{7} is incorrect.
  • B. x=127x = \frac{12}{7} is incorrect.
  • C. x=−4x = -4 is correct.
  • D. x=4x = 4 is incorrect.

Discussion

  • What is the first step in solving a linear equation?
  • How do you eliminate fractions in a linear equation?
  • What is the importance of isolating the variable x in a linear equation?
  • How do you solve for x in a linear equation?

Additional Resources

  • Khan Academy: Solving Linear Equations
  • Mathway: Solving Linear Equations
  • Wolfram Alpha: Solving Linear Equations

Final Thoughts

Solving linear equations is a fundamental skill in mathematics, and it requires a systematic approach. By following the steps outlined in this article, you can solve linear equations with confidence. Remember to eliminate fractions, isolate the variable x, and solve for x. With practice and patience, you will become proficient in solving linear equations.

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Q&A: Solving Linear Equations

Q: What is a linear equation?

A: A linear equation is an equation in which the highest power of the variable(s) is 1. It is a simple equation that can be solved using basic algebraic operations.

Q: What are the steps to solve a linear equation?

A: The steps to solve a linear equation are:

  1. Eliminate fractions by multiplying both sides of the equation by the least common multiple (LCM) of the denominators.
  2. Isolate the variable x by adding or subtracting the same value to both sides of the equation.
  3. Solve for x by dividing both sides of the equation by the coefficient of x.

Q: How do I eliminate fractions in a linear equation?

A: To eliminate fractions in a linear equation, multiply both sides of the equation by the least common multiple (LCM) of the denominators.

Q: What is the importance of isolating the variable x in a linear equation?

A: Isolating the variable x in a linear equation is important because it allows you to solve for x. By isolating x, you can determine the value of x that satisfies the equation.

Q: How do I solve for x in a linear equation?

A: To solve for x in a linear equation, divide both sides of the equation by the coefficient of x.

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

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

  • Not eliminating fractions
  • Not isolating the variable x
  • Not solving for x
  • Making arithmetic errors

Q: How can I practice solving linear equations?

A: You can practice solving linear equations by:

  • Using online resources such as Khan Academy or Mathway
  • Working with a tutor or teacher
  • Practicing with sample problems
  • Creating your own problems to solve

Q: What are some real-world applications of linear equations?

A: Linear equations have many real-world applications, including:

  • Physics: to describe the motion of objects
  • Engineering: to design and optimize systems
  • Economics: to model economic systems
  • Computer Science: to solve problems in computer programming

Conclusion

Solving linear equations is a fundamental skill in mathematics, and it requires a systematic approach. By following the steps outlined in this article, you can solve linear equations with confidence. Remember to eliminate fractions, isolate the variable x, and solve for x. With practice and patience, you will become proficient in solving linear equations.

Additional Resources

  • Khan Academy: Solving Linear Equations
  • Mathway: Solving Linear Equations
  • Wolfram Alpha: Solving Linear Equations

Final Thoughts

Solving linear equations is a crucial skill in mathematics, and it has many real-world applications. By practicing and mastering this skill, you can improve your problem-solving abilities and become more confident in your mathematical abilities.