Solve For { X $}$ In The Equation: ${ X - 2 \frac{5}{7} = \frac{2}{3} }$
Introduction to Solving Equations
Solving equations is a fundamental concept in mathematics, and it is essential to understand how to isolate variables to find their values. In this article, we will focus on solving a specific equation involving fractions and decimals. We will use algebraic techniques to isolate the variable and find its value.
Understanding the Equation
The given equation is: ${ x - 2 \frac{5}{7} = \frac{2}{3} }$
To solve for , we need to isolate it on one side of the equation. The first step is to simplify the left-hand side of the equation by converting the mixed number to an improper fraction.
Converting Mixed Numbers to Improper Fractions
A mixed number is a combination of a whole number and a fraction. To convert a mixed number to an improper fraction, we multiply the whole number by the denominator and add the numerator. Then, we write the result as an improper fraction with the same denominator.
In this case, the mixed number is . To convert it to an improper fraction, we multiply the whole number by the denominator and add the numerator:
So, the improper fraction is .
Simplifying the Equation
Now that we have converted the mixed number to an improper fraction, we can simplify the equation:
Adding Fractions with Different Denominators
To add fractions with different denominators, we need to find a common denominator. In this case, the least common multiple (LCM) of 7 and 3 is 21.
We can rewrite the fractions with the common denominator:
{ x - \frac{19 \times 3}{7 \times 3} = \frac{2 \times 7}{3 \times 7} }$ ${ x - \frac{57}{21} = \frac{14}{21} }$ ## Adding Fractions with the Same Denominator Now that we have the fractions with the same denominator, we can add them: ${ x = \frac{14}{21} + \frac{57}{21} }$ ${ x = \frac{71}{21} }$ ## Conclusion In this article, we solved the equation ${ x - 2 \frac{5}{7} = \frac{2}{3} }$. We converted the mixed number to an improper fraction, simplified the equation, and added fractions with different denominators. Finally, we isolated the variable $x$ and found its value. ## Tips and Tricks * When solving equations involving fractions, it is essential to find a common denominator to add or subtract fractions. * To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. * When adding fractions with the same denominator, simply add the numerators and keep the denominator the same. ## Real-World Applications Solving equations involving fractions has many real-world applications, such as: * Finance: When calculating interest rates or investment returns, fractions are often used to represent the interest or return on investment. * Science: In physics and chemistry, fractions are used to represent measurements and calculations. * Engineering: In engineering, fractions are used to represent dimensions and calculations. ## Final Thoughts Solving equations involving fractions requires a deep understanding of algebraic techniques and fraction operations. By following the steps outlined in this article, you can solve equations involving fractions and apply the concepts to real-world problems. Remember to always find a common denominator when adding or subtracting fractions, and to convert mixed numbers to improper fractions when necessary.<br/> # Frequently Asked Questions (FAQs) on Solving Equations Involving Fractions ## Introduction Solving equations involving fractions can be a challenging task, but with the right techniques and strategies, it can be made easier. In this article, we will answer some frequently asked questions (FAQs) on solving equations involving fractions. ## Q: What is the first step in solving an equation involving fractions? A: The first step in solving an equation involving fractions is to simplify the equation by converting any mixed numbers to improper fractions. ## Q: How do I convert a mixed number to an improper fraction? A: To convert a mixed number to an improper fraction, multiply the whole number by the denominator and add the numerator. Then, write the result as an improper fraction with the same denominator. ## Q: What is the least common multiple (LCM) of two numbers? A: The least common multiple (LCM) of two numbers is the smallest number that is a multiple of both numbers. ## Q: How do I find the LCM of two numbers? A: To find the LCM of two numbers, list the multiples of each number and find the smallest number that appears in both lists. ## Q: What is the difference between adding and subtracting fractions with the same denominator? A: When adding fractions with the same denominator, simply add the numerators and keep the denominator the same. When subtracting fractions with the same denominator, subtract the numerators and keep the denominator the same. ## Q: How do I add fractions with different denominators? A: To add fractions with different denominators, find a common denominator and rewrite the fractions with the common denominator. Then, add the fractions. ## Q: What is the final step in solving an equation involving fractions? A: The final step in solving an equation involving fractions is to isolate the variable by adding or subtracting the same value to both sides of the equation. ## Q: Can I use a calculator to solve equations involving fractions? A: Yes, you can use a calculator to solve equations involving fractions. However, it is essential to understand the underlying math concepts and techniques to ensure that you are using the calculator correctly. ## Q: What are some real-world applications of solving equations involving fractions? A: Solving equations involving fractions has many real-world applications, such as finance, science, and engineering. In finance, fractions are used to represent interest rates and investment returns. In science, fractions are used to represent measurements and calculations. In engineering, fractions are used to represent dimensions and calculations. ## Q: Can I use a graphing calculator to solve equations involving fractions? A: Yes, you can use a graphing calculator to solve equations involving fractions. Graphing calculators can help you visualize the equation and find the solution. ## Q: What are some common mistakes to avoid when solving equations involving fractions? A: Some common mistakes to avoid when solving equations involving fractions include: * Not simplifying the equation before solving * Not finding a common denominator when adding or subtracting fractions * Not isolating the variable by adding or subtracting the same value to both sides of the equation * Not using a calculator correctly ## Conclusion Solving equations involving fractions requires a deep understanding of algebraic techniques and fraction operations. By following the steps outlined in this article and avoiding common mistakes, you can solve equations involving fractions and apply the concepts to real-world problems. ## Tips and Tricks * Always simplify the equation before solving * Find a common denominator when adding or subtracting fractions * Isolate the variable by adding or subtracting the same value to both sides of the equation * Use a calculator correctly * Practice, practice, practice! ## Final Thoughts Solving equations involving fractions is a challenging task, but with the right techniques and strategies, it can be made easier. By following the steps outlined in this article and avoiding common mistakes, you can solve equations involving fractions and apply the concepts to real-world problems. Remember to always simplify the equation before solving, find a common denominator when adding or subtracting fractions, and isolate the variable by adding or subtracting the same value to both sides of the equation.</span></p>