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Solving Linear Equations with Fractions: A Step-by-Step Guide

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Solving Linear Equations with Fractions

Solving linear equations is a fundamental skill in algebra, and it often involves working with fractions. While fractions can seem daunting, the process of solving linear equations with fractions is quite straightforward. This guide will provide a step-by-step approach to tackling these equations with confidence.

Understanding the Concept

A linear equation is an equation where the highest power of the variable is 1. For example, 2x + 5 = 11 is a linear equation. When fractions are involved, it means the variable or a constant term is divided by a number. For instance, x/2 + 3 = 7 is a linear equation with a fraction.

Steps to Solve Linear Equations with Fractions

To solve linear equations with fractions, follow these steps:

  1. Find the Least Common Denominator (LCD): Identify the denominators of all the fractions in the equation. The LCD is the smallest number that is divisible by all the denominators. For example, in the equation x/2 + 3/4 = 5/6, the LCD is 12.
  2. Multiply Both Sides by the LCD: Multiply both sides of the equation by the LCD. This will eliminate the fractions, as the LCD will cancel out the denominators. In our example, multiplying both sides by 12 gives us: 6x + 9 = 10.
  3. Solve for the Variable: Now that the fractions are gone, you can solve the equation for the variable using standard algebraic techniques. In our example, we would subtract 9 from both sides, then divide by 6: 6x = 1, x = 1/6.

Example

Let's solve the equation 2x/3 - 1/4 = 5/6.

  1. Find the LCD: The LCD of 3, 4, and 6 is 12.
  2. Multiply by the LCD: Multiplying both sides by 12 gives us: 8x - 3 = 10.
  3. Solve for x: Adding 3 to both sides, we get: 8x = 13. Dividing both sides by 8, we get: x = 13/8.

Tips for Solving Linear Equations with Fractions

  • Simplify Fractions First: If possible, simplify any fractions in the equation before finding the LCD. This can make the calculations easier.
  • Check Your Answer: After solving the equation, always substitute your solution back into the original equation to verify that it is correct.
  • Practice Makes Perfect: The more you practice solving linear equations with fractions, the more comfortable and confident you will become.

Conclusion

Solving linear equations with fractions may seem challenging at first, but by following the steps outlined above, you can master this skill. Remember to find the LCD, multiply both sides by it, and then solve for the variable. With practice, you will be able to tackle these equations with ease.

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