Solve For \[$ X \$\].$\[ \frac{2}{3} X = \frac{8}{9} \\]

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Introduction

Solving for xx in an equation is a fundamental concept in mathematics, and it is essential to understand how to isolate the variable in a given equation. In this article, we will focus on solving for xx in a simple linear equation, specifically the equation 23x=89\frac{2}{3} x = \frac{8}{9}.

Understanding the Equation

The given equation is a linear equation, which means it is an equation in which the highest power of the variable xx is 1. The equation is 23x=89\frac{2}{3} x = \frac{8}{9}. To solve for xx, we need to isolate the variable xx on one side of the equation.

Isolating the Variable

To isolate the variable xx, we need to get rid of the coefficient 23\frac{2}{3} that is multiplied by xx. We can do this by multiplying both sides of the equation by the reciprocal of the coefficient, which is 32\frac{3}{2}.

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

We multiply both sides of the equation by 32\frac{3}{2} to get rid of the coefficient 23\frac{2}{3}.

32โ‹…23x=32โ‹…89\frac{3}{2} \cdot \frac{2}{3} x = \frac{3}{2} \cdot \frac{8}{9}

Step 1 Simplified

Simplifying the left-hand side of the equation, we get:

x=32โ‹…89x = \frac{3}{2} \cdot \frac{8}{9}

Step 2: Simplify the Right-Hand Side

To simplify the right-hand side of the equation, we multiply the numerators and denominators separately.

x=3โ‹…82โ‹…9x = \frac{3 \cdot 8}{2 \cdot 9}

Step 2 Simplified

Simplifying the right-hand side of the equation, we get:

x=2418x = \frac{24}{18}

Step 3: Simplify the Fraction

To simplify the fraction, we divide both the numerator and denominator by their greatest common divisor, which is 6.

x=24รท618รท6x = \frac{24 \div 6}{18 \div 6}

Step 3 Simplified

Simplifying the fraction, we get:

x=43x = \frac{4}{3}

Conclusion

In this article, we solved for xx in the equation 23x=89\frac{2}{3} x = \frac{8}{9}. We isolated the variable xx by multiplying both sides of the equation by the reciprocal of the coefficient, and then simplified the resulting fraction to get the final answer, which is x=43x = \frac{4}{3}.

Tips and Tricks

  • When solving for xx, make sure to isolate the variable on one side of the equation.
  • Use the reciprocal of the coefficient to get rid of the coefficient.
  • Simplify the resulting fraction by dividing both the numerator and denominator by their greatest common divisor.

Real-World Applications

Solving for xx has many real-world applications, such as:

  • Calculating the cost of an item based on its price and the number of items purchased.
  • Determining the amount of time it takes to complete a task based on the rate of work and the amount of work done.
  • Finding the area of a rectangle based on its length and width.

Common Mistakes

  • Failing to isolate the variable on one side of the equation.
  • Not using the reciprocal of the coefficient to get rid of the coefficient.
  • Not simplifying the resulting fraction.

Conclusion

Solving for xx is a fundamental concept in mathematics, and it has many real-world applications. By following the steps outlined in this article, you can solve for xx in any linear equation. Remember to isolate the variable on one side of the equation, use the reciprocal of the coefficient, and simplify the resulting fraction.

Introduction

In our previous article, we solved for xx in the equation 23x=89\frac{2}{3} x = \frac{8}{9}. In this article, we will answer some frequently asked questions about solving for xx.

Q: What is solving for xx?

A: Solving for xx is the process of isolating the variable xx on one side of an equation. This means that we need to get rid of any coefficients or constants that are multiplied by xx.

Q: Why is it important to isolate the variable xx?

A: Isolating the variable xx is important because it allows us to find the value of xx that satisfies the equation. If we don't isolate the variable, we won't be able to find the value of xx.

Q: How do I isolate the variable xx?

A: To isolate the variable xx, you need to get rid of any coefficients or constants that are multiplied by xx. You can do this by multiplying both sides of the equation by the reciprocal of the coefficient.

Q: What is the reciprocal of a coefficient?

A: The reciprocal of a coefficient is the number that you need to multiply by the coefficient to get 1. For example, the reciprocal of 23\frac{2}{3} is 32\frac{3}{2}.

Q: How do I simplify a fraction?

A: To simplify a fraction, you need to divide both the numerator and denominator by their greatest common divisor (GCD). The GCD is the largest number that divides both the numerator and denominator without leaving a remainder.

Q: What is the greatest common divisor (GCD)?

A: The greatest common divisor (GCD) is the largest number that divides both the numerator and denominator without leaving a remainder. For example, the GCD of 24 and 18 is 6.

Q: How do I find the GCD of two numbers?

A: To find the GCD of two numbers, you can use the following steps:

  1. List the factors of each number.
  2. Identify the common factors.
  3. Choose the largest common factor.

Q: What are some common mistakes to avoid when solving for xx?

A: Some common mistakes to avoid when solving for xx include:

  • Failing to isolate the variable xx on one side of the equation.
  • Not using the reciprocal of the coefficient to get rid of the coefficient.
  • Not simplifying the resulting fraction.

Q: How do I check my answer?

A: To check your answer, you can plug the value of xx back into the original equation and see if it is true. If it is true, then your answer is correct.

Q: What are some real-world applications of solving for xx?

A: Solving for xx has many real-world applications, such as:

  • Calculating the cost of an item based on its price and the number of items purchased.
  • Determining the amount of time it takes to complete a task based on the rate of work and the amount of work done.
  • Finding the area of a rectangle based on its length and width.

Conclusion

Solving for xx is a fundamental concept in mathematics, and it has many real-world applications. By following the steps outlined in this article, you can solve for xx in any linear equation. Remember to isolate the variable xx on one side of the equation, use the reciprocal of the coefficient, and simplify the resulting fraction.

Additional Resources

  • For more information on solving for xx, check out our previous article on the topic.
  • For practice problems and exercises, try using online resources such as Khan Academy or Mathway.
  • For more advanced topics in mathematics, try checking out online resources such as MIT OpenCourseWare or Coursera.

Frequently Asked Questions

  • Q: What is the difference between solving for xx and solving for yy? A: Solving for xx and solving for yy are both used to isolate a variable on one side of an equation. The only difference is that xx is typically used as the variable in linear equations, while yy is typically used as the variable in quadratic equations.
  • Q: How do I solve for xx in a quadratic equation? A: To solve for xx in a quadratic equation, you need to use the quadratic formula: x=โˆ’bยฑb2โˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.
  • Q: What is the quadratic formula? A: The quadratic formula is a mathematical formula that is used to solve quadratic equations. It is given by: x=โˆ’bยฑb2โˆ’4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}.

Conclusion

Solving for xx is a fundamental concept in mathematics, and it has many real-world applications. By following the steps outlined in this article, you can solve for xx in any linear equation. Remember to isolate the variable xx on one side of the equation, use the reciprocal of the coefficient, and simplify the resulting fraction.