Solve: { \frac{2}{3}x = 8$}$A. 4 B. 16 C. 12 D. 24 E. { \frac{16}{3}$}$

by ADMIN 76 views

Introduction

In this article, we will be solving a linear equation involving fractions. The equation is {\frac{2}{3}x = 8$}$. Our goal is to isolate the variable x and find its value. We will use the properties of fractions and algebraic operations to solve this equation.

Step 1: Multiply Both Sides by the Reciprocal of the Fraction

To get rid of the fraction, we need to multiply both sides of the equation by the reciprocal of {\frac{2}{3}$}$, which is {\frac{3}{2}$}$. This will eliminate the fraction and allow us to solve for x.

{\frac{2}{3}x = 8$}$

Multiply both sides by {\frac{3}{2}$}$:

{\frac{3}{2} \times \frac{2}{3}x = \frac{3}{2} \times 8$}$

Simplify the equation:

{x = 12$}$

Step 2: Check the Solution

To verify that our solution is correct, we can plug it back into the original equation and check if it is true.

{\frac{2}{3}x = 8$}$

Substitute x = 12:

{\frac{2}{3} \times 12 = 8$}$

Simplify the equation:

${8 = 8\$}

Since the equation is true, our solution is correct.

Conclusion

In this article, we solved the linear equation {\frac{2}{3}x = 8$}$ by multiplying both sides by the reciprocal of the fraction. We found that the value of x is 12. We also verified our solution by plugging it back into the original equation.

Final Answer

The final answer is C. 12.

Why is this the Correct Answer?

This is the correct answer because we solved the equation correctly by multiplying both sides by the reciprocal of the fraction. We also verified our solution by plugging it back into the original equation, which showed that the equation is true.

What is the Importance of Solving Linear Equations?

Solving linear equations is an important skill in mathematics because it allows us to solve problems that involve variables and constants. Linear equations are used in a wide range of applications, including physics, engineering, and economics. By solving linear equations, we can model real-world problems and make predictions about the behavior of systems.

How to Solve Linear Equations with Fractions

To solve linear equations with fractions, we need to follow these steps:

  1. Multiply both sides of the equation by the reciprocal of the fraction.
  2. Simplify the equation.
  3. Check the solution by plugging it back into the original equation.

By following these steps, we can solve linear equations with fractions and find the value of the variable.

Common Mistakes to Avoid

When solving linear equations with fractions, there are several common mistakes to avoid:

  1. Not multiplying both sides of the equation by the reciprocal of the fraction.
  2. Not simplifying the equation.
  3. Not checking the solution by plugging it back into the original equation.

By avoiding these mistakes, we can ensure that our solutions are correct and accurate.

Real-World Applications

Linear equations with fractions have many real-world applications, including:

  1. Physics: Linear equations are used to model the motion of objects and the behavior of physical systems.
  2. Engineering: Linear equations are used to design and optimize systems, such as bridges and buildings.
  3. Economics: Linear equations are used to model economic systems and make predictions about the behavior of markets.

By solving linear equations with fractions, we can model real-world problems and make predictions about the behavior of systems.

Conclusion

In this article, we solved the linear equation {\frac{2}{3}x = 8$}$ by multiplying both sides by the reciprocal of the fraction. We found that the value of x is 12. We also verified our solution by plugging it back into the original equation. By following the steps outlined in this article, we can solve linear equations with fractions and find the value of the variable.

Introduction

In our previous article, we solved the linear equation {\frac{2}{3}x = 8$}$ by multiplying both sides by the reciprocal of the fraction. In this article, we will answer some common questions that students may have when solving linear equations with fractions.

Q: What is the reciprocal of a fraction?

A: The reciprocal of a fraction is obtained by swapping the numerator and the denominator. For example, the reciprocal of {\frac{2}{3}$}$ is {\frac{3}{2}$}$.

Q: Why do we need to multiply both sides of the equation by the reciprocal of the fraction?

A: We need to multiply both sides of the equation by the reciprocal of the fraction to eliminate the fraction and solve for the variable. This is because the reciprocal of a fraction is the number that, when multiplied by the fraction, gives 1.

Q: How do we know which side of the equation to multiply by the reciprocal of the fraction?

A: We multiply the side of the equation that contains the fraction by the reciprocal of the fraction. This is because we want to eliminate the fraction and solve for the variable.

Q: What if the equation has multiple fractions?

A: If the equation has multiple fractions, we need to multiply both sides of the equation by the reciprocal of each fraction, one at a time. This will eliminate each fraction and allow us to solve for the variable.

Q: Can we simplify the equation before multiplying by the reciprocal of the fraction?

A: Yes, we can simplify the equation before multiplying by the reciprocal of the fraction. This will make it easier to solve for the variable.

Q: How do we check the solution?

A: We check the solution by plugging it back into the original equation and verifying that it is true.

Q: What if the solution is not an integer?

A: If the solution is not an integer, it means that the equation has a fractional solution. This is perfectly valid and can be expressed as a fraction.

Q: Can we use other methods to solve linear equations with fractions?

A: Yes, we can use other methods to solve linear equations with fractions, such as using algebraic manipulations or using a calculator. However, multiplying by the reciprocal of the fraction is a simple and effective method.

Q: Why is it important to solve linear equations with fractions?

A: Solving linear equations with fractions is an important skill in mathematics because it allows us to model real-world problems and make predictions about the behavior of systems.

Q: Can we apply this method to other types of equations?

A: Yes, we can apply this method to other types of equations, such as quadratic equations or polynomial equations. However, the method may need to be modified to accommodate the specific type of equation.

Conclusion

In this article, we answered some common questions that students may have when solving linear equations with fractions. We also provided examples and explanations to help illustrate the concepts. By following the steps outlined in this article, students can develop a deeper understanding of linear equations with fractions and improve their problem-solving skills.

Final Tips

  • Always multiply both sides of the equation by the reciprocal of the fraction to eliminate the fraction.
  • Simplify the equation before multiplying by the reciprocal of the fraction.
  • Check the solution by plugging it back into the original equation.
  • Be careful when working with fractions and make sure to simplify the equation before solving for the variable.

By following these tips and practicing with examples, students can become proficient in solving linear equations with fractions and apply this skill to a wide range of real-world problems.