Solve For $x$. 3 X − 8 = − 21 4 \frac{3x}{-8} = \frac{-21}{4} − 8 3 X ​ = 4 − 21 ​ Enter Your Answer In The Box. X = X = X = [ ]

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Introduction

Solving equations is a fundamental concept in mathematics, and it's essential to understand how to isolate variables to find their values. In this article, we will focus on solving a specific type of equation, which involves fractions. We will use the given equation 3x−8=−214\frac{3x}{-8} = \frac{-21}{4} as an example and walk through the steps to solve for xx.

Understanding the Equation

The given equation is a rational equation, which means it involves fractions. The equation is 3x−8=−214\frac{3x}{-8} = \frac{-21}{4}. Our goal is to solve for xx, which means we need to isolate the variable xx on one side of the equation.

Step 1: Multiply Both Sides by the Least Common Multiple (LCM)

To eliminate the fractions, we need to multiply both sides of the equation by the least common multiple (LCM) of the denominators. In this case, the denominators are −8-8 and 44. The LCM of −8-8 and 44 is −8×4=−32-8 \times 4 = -32.

Multiplying both sides by -32:
$\frac{3x}{-8} \times -32 = \frac{-21}{4} \times -32$

Step 2: Simplify the Equation

After multiplying both sides by −32-32, we can simplify the equation.

Simplifying the equation:
$-3x \times 4 = -21 \times 8$
$-12x = -168$

Step 3: Divide Both Sides by the Coefficient

To isolate the variable xx, we need to divide both sides of the equation by the coefficient of xx. In this case, the coefficient of xx is −12-12.

Dividing both sides by -12:
$\frac{-12x}{-12} = \frac{-168}{-12}$
$x = 14$

Conclusion

In this article, we solved the equation 3x−8=−214\frac{3x}{-8} = \frac{-21}{4} for xx. We used the steps of multiplying both sides by the least common multiple (LCM) of the denominators, simplifying the equation, and dividing both sides by the coefficient of xx. The final solution is x=14x = 14.

Tips and Tricks

  • When solving rational equations, it's essential to multiply both sides by the least common multiple (LCM) of the denominators to eliminate the fractions.
  • Make sure to simplify the equation after multiplying both sides by the LCM.
  • When dividing both sides by the coefficient, make sure to divide both sides by the coefficient, not just the numerator.

Practice Problems

  • Solve the equation 2x−6=−153\frac{2x}{-6} = \frac{-15}{3} for xx.
  • Solve the equation x−4=−128\frac{x}{-4} = \frac{-12}{8} for xx.

Real-World Applications

Solving rational equations has many real-world applications, such as:

  • Finance: Solving rational equations can help you calculate interest rates, investment returns, and other financial metrics.
  • Science: Solving rational equations can help you calculate rates of change, acceleration, and other scientific metrics.
  • Engineering: Solving rational equations can help you design and optimize systems, such as electrical circuits and mechanical systems.

Final Thoughts

Solving rational equations is a fundamental concept in mathematics, and it's essential to understand how to isolate variables to find their values. By following the steps outlined in this article, you can solve rational equations and apply them to real-world problems. Remember to multiply both sides by the least common multiple (LCM) of the denominators, simplify the equation, and divide both sides by the coefficient of xx. With practice and patience, you can become proficient in solving rational equations and apply them to a wide range of problems.

Introduction

In our previous article, we solved the equation 3x−8=−214\frac{3x}{-8} = \frac{-21}{4} for xx. We used the steps of multiplying both sides by the least common multiple (LCM) of the denominators, simplifying the equation, and dividing both sides by the coefficient of xx. In this article, we will answer some frequently asked questions (FAQs) about solving rational equations.

Q&A

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. For example, the LCM of 4 and 6 is 12.

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

A: To find the LCM of two numbers, you can list the multiples of each number and find the smallest number that appears in both lists. Alternatively, you can use the formula: LCM(a, b) = (a × b) / GCD(a, b), where GCD is the greatest common divisor.

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

A: The greatest common divisor (GCD) of two numbers is the largest number that divides both numbers without leaving a remainder. For example, the GCD of 12 and 18 is 6.

Q: How do I simplify a rational equation?

A: To simplify a rational equation, you can multiply both sides by the least common multiple (LCM) of the denominators, and then simplify the resulting equation.

Q: What is the difference between a rational equation and a rational expression?

A: A rational equation is an equation that involves rational expressions, while a rational expression is a fraction that involves variables or constants.

Q: How do I solve a rational equation with a variable in the denominator?

A: To solve a rational equation with a variable in the denominator, you can multiply both sides by the least common multiple (LCM) of the denominators, and then simplify the resulting equation.

Q: What is the final answer to the equation 3x−8=−214\frac{3x}{-8} = \frac{-21}{4}?

A: The final answer to the equation 3x−8=−214\frac{3x}{-8} = \frac{-21}{4} is x=14x = 14.

Tips and Tricks

  • When solving rational equations, it's essential to multiply both sides by the least common multiple (LCM) of the denominators to eliminate the fractions.
  • Make sure to simplify the equation after multiplying both sides by the LCM.
  • When dividing both sides by the coefficient, make sure to divide both sides by the coefficient, not just the numerator.

Practice Problems

  • Solve the equation 2x−6=−153\frac{2x}{-6} = \frac{-15}{3} for xx.
  • Solve the equation x−4=−128\frac{x}{-4} = \frac{-12}{8} for xx.

Real-World Applications

Solving rational equations has many real-world applications, such as:

  • Finance: Solving rational equations can help you calculate interest rates, investment returns, and other financial metrics.
  • Science: Solving rational equations can help you calculate rates of change, acceleration, and other scientific metrics.
  • Engineering: Solving rational equations can help you design and optimize systems, such as electrical circuits and mechanical systems.

Final Thoughts

Solving rational equations is a fundamental concept in mathematics, and it's essential to understand how to isolate variables to find their values. By following the steps outlined in this article, you can solve rational equations and apply them to real-world problems. Remember to multiply both sides by the least common multiple (LCM) of the denominators, simplify the equation, and divide both sides by the coefficient of xx. With practice and patience, you can become proficient in solving rational equations and apply them to a wide range of problems.

Additional Resources

  • Khan Academy: Rational Equations
  • Mathway: Rational Equations
  • Wolfram Alpha: Rational Equations

Conclusion

In this article, we answered some frequently asked questions (FAQs) about solving rational equations. We covered topics such as the least common multiple (LCM), greatest common divisor (GCD), simplifying rational equations, and solving rational equations with variables in the denominator. We also provided practice problems and real-world applications of solving rational equations. With practice and patience, you can become proficient in solving rational equations and apply them to a wide range of problems.