Solve For { X $} . . . { 5x - \frac{3}{2} = \frac{5}{2}x + \frac{15}{4} \}

by ADMIN 75 views

Introduction to Solving Linear Equations

Solving linear equations is a fundamental concept in mathematics, and it is essential to understand how to isolate the variable in a linear equation. In this article, we will focus on solving a linear equation that involves fractions. We will use the given equation 5xβˆ’32=52x+1545x - \frac{3}{2} = \frac{5}{2}x + \frac{15}{4} as an example to demonstrate the steps involved in solving for xx.

Understanding the Equation

The given equation is a linear equation that involves fractions. To solve for xx, we need to isolate the variable xx on one side of the equation. The equation can be rewritten as 5xβˆ’52x=154+325x - \frac{5}{2}x = \frac{15}{4} + \frac{3}{2}.

Combining Like Terms

To simplify the equation, we need to combine like terms. The left-hand side of the equation can be simplified by combining the terms involving xx. We can rewrite the equation as x(5βˆ’52)=154+32x(5 - \frac{5}{2}) = \frac{15}{4} + \frac{3}{2}.

Simplifying the Equation

To simplify the equation further, we need to evaluate the expressions inside the parentheses. We can rewrite the equation as x(102βˆ’52)=154+32x(\frac{10}{2} - \frac{5}{2}) = \frac{15}{4} + \frac{3}{2}.

Evaluating Expressions

To evaluate the expressions inside the parentheses, we need to find a common denominator. The common denominator for the fractions inside the parentheses is 2. We can rewrite the equation as x(52)=154+32x(\frac{5}{2}) = \frac{15}{4} + \frac{3}{2}.

Adding Fractions

To add the fractions on the right-hand side of the equation, we need to find a common denominator. The common denominator for the fractions is 4. We can rewrite the equation as x(52)=154+64x(\frac{5}{2}) = \frac{15}{4} + \frac{6}{4}.

Simplifying the Right-Hand Side

To simplify the right-hand side of the equation, we can add the fractions. We can rewrite the equation as x(52)=214x(\frac{5}{2}) = \frac{21}{4}.

Isolating the Variable

To isolate the variable xx, we need to divide both sides of the equation by the coefficient of xx. The coefficient of xx is 52\frac{5}{2}. We can rewrite the equation as x=214Γ·52x = \frac{21}{4} \div \frac{5}{2}.

Dividing Fractions

To divide fractions, we need to invert the second fraction and multiply. We can rewrite the equation as x=214Γ—25x = \frac{21}{4} \times \frac{2}{5}.

Simplifying the Expression

To simplify the expression, we can multiply the numerators and denominators. We can rewrite the equation as x=21Γ—24Γ—5x = \frac{21 \times 2}{4 \times 5}.

Evaluating the Expression

To evaluate the expression, we need to multiply the numerators and denominators. We can rewrite the equation as x=4220x = \frac{42}{20}.

Simplifying the Fraction

To simplify the fraction, we need to find the greatest common divisor (GCD) of the numerator and denominator. The GCD of 42 and 20 is 2. We can rewrite the equation as x=2110x = \frac{21}{10}.

Conclusion

In this article, we have demonstrated the steps involved in solving a linear equation that involves fractions. We have used the given equation 5xβˆ’32=52x+1545x - \frac{3}{2} = \frac{5}{2}x + \frac{15}{4} as an example to illustrate the process of solving for xx. We have shown how to combine like terms, simplify the equation, evaluate expressions, add fractions, isolate the variable, divide fractions, simplify the expression, and evaluate the expression. By following these steps, we can solve linear equations that involve fractions.

Final Answer

The final answer to the equation 5xβˆ’32=52x+1545x - \frac{3}{2} = \frac{5}{2}x + \frac{15}{4} is x=2110x = \frac{21}{10}.

Tips and Tricks

  • When solving linear equations that involve fractions, it is essential to find a common denominator to add or subtract fractions.
  • When dividing fractions, it is essential to invert the second fraction and multiply.
  • When simplifying fractions, it is essential to find the greatest common divisor (GCD) of the numerator and denominator.
  • When solving linear equations, it is essential to isolate the variable on one side of the equation.

Common Mistakes

  • Not finding a common denominator when adding or subtracting fractions.
  • Not inverting the second fraction when dividing fractions.
  • Not simplifying fractions by finding the greatest common divisor (GCD) of the numerator and denominator.
  • Not isolating the variable on one side of the equation.

Real-World Applications

Solving linear equations that involve fractions has numerous real-world applications. For example, in finance, linear equations are used to calculate interest rates and investment returns. In science, linear equations are used to model population growth and chemical reactions. In engineering, linear equations are used to design and optimize systems.

Future Research

Future research in solving linear equations that involve fractions could focus on developing new algorithms and techniques for solving these types of equations. Additionally, research could focus on applying linear equations to real-world problems in fields such as finance, science, and engineering.

Conclusion

In conclusion, solving linear equations that involve fractions is a fundamental concept in mathematics. By following the steps outlined in this article, we can solve linear equations that involve fractions. The final answer to the equation 5xβˆ’32=52x+1545x - \frac{3}{2} = \frac{5}{2}x + \frac{15}{4} is x=2110x = \frac{21}{10}.

Introduction

Solving linear equations with fractions can be a challenging task, but with the right approach, it can be made easier. In this article, we will provide a Q&A section to help you understand the concepts and techniques involved in solving linear equations with fractions.

Q1: What is a linear equation with fractions?

A1: A linear equation with fractions is an equation that involves fractions and can be written in the form ax + b = cx + d, where a, b, c, and d are constants, and x is the variable.

Q2: How do I simplify a linear equation with fractions?

A2: To simplify a linear equation with fractions, you need to find a common denominator for the fractions and then combine the fractions. You can also use the distributive property to simplify the equation.

Q3: How do I add or subtract fractions in a linear equation?

A3: To add or subtract fractions in a linear equation, you need to find a common denominator for the fractions. Once you have a common denominator, you can add or subtract the numerators.

Q4: How do I divide fractions in a linear equation?

A4: To divide fractions in a linear equation, you need to invert the second fraction and multiply. This means that if you have a fraction a/b divided by a fraction c/d, you can rewrite it as a/b Γ— d/c.

Q5: How do I isolate the variable in a linear equation with fractions?

A5: To isolate the variable in a linear equation with fractions, you need to get all the terms with the variable on one side of the equation and the constant terms on the other side. You can do this by adding or subtracting fractions and then dividing or multiplying by a constant.

Q6: What are some common mistakes to avoid when solving linear equations with fractions?

A6: Some common mistakes to avoid when solving linear equations with fractions include not finding a common denominator, not inverting the second fraction when dividing, and not simplifying fractions by finding the greatest common divisor (GCD) of the numerator and denominator.

Q7: How do I check my answer when solving a linear equation with fractions?

A7: To check your answer when solving a linear equation with fractions, you need to plug your solution back into the original equation and make sure it is true. You can also use a calculator to check your answer.

Q8: What are some real-world applications of solving linear equations with fractions?

A8: Solving linear equations with fractions has numerous real-world applications, including finance, science, and engineering. For example, in finance, linear equations are used to calculate interest rates and investment returns. In science, linear equations are used to model population growth and chemical reactions. In engineering, linear equations are used to design and optimize systems.

Q9: What are some tips for solving linear equations with fractions?

A9: Some tips for solving linear equations with fractions include finding a common denominator, inverting the second fraction when dividing, and simplifying fractions by finding the greatest common divisor (GCD) of the numerator and denominator.

Q10: How do I practice solving linear equations with fractions?

A10: To practice solving linear equations with fractions, you can start by working through some examples and then try solving some problems on your own. You can also use online resources, such as video tutorials and practice problems, to help you improve your skills.

Conclusion

Solving linear equations with fractions can be a challenging task, but with the right approach, it can be made easier. By following the steps outlined in this article and practicing regularly, you can become proficient in solving linear equations with fractions. Remember to find a common denominator, invert the second fraction when dividing, and simplify fractions by finding the greatest common divisor (GCD) of the numerator and denominator.