Solving Linear Equations Involving Fractions and Decimals: A Comprehensive Guide

Linear equations with fractions and decimals can seem daunting, but by following a few simple steps, you can solve them with confidence. This guide will walk you through the process of solving linear equations involving fractions and decimals using the least common denominator (LCD) and least common multiple (LCM) methods.

Understanding Fractions and Decimals in Linear Equations

Linear equations with fractions and decimals often appear in various real-world applications, such as in physics, engineering, and finance. Understanding how to solve these types of equations is crucial for many fields and everyday problem-solving. In this article, we will explore the step-by-step process to solve linear equations involving fractions and decimals.

Step 1: Eliminate Fractions

Eliminating fractions makes the calculations easier and the solution more straightforward. The first step involves finding the least common denominator (LCD) of all the fractions in the equation and multiplying each term by the LCD.

Example Equation

Let's solve the following equation as an example:

frac{2}{3}x - frac{1}{4} frac{5}{6}

Step 1: Eliminate Fractions

To eliminate the fractions, we need to multiply every term in the equation by the least common denominator (LCD) of the fractions involved. In this case, the denominators are 3, 4, and 6. The LCD for these numbers is 12.

Multiplying every term by 12:

12 left( frac{2}{3}x right) - 12 left( frac{1}{4} right) 12 left( frac{5}{6} right)

This simplifies to:

8x - 3 10

Step 2: Isolate the Variable

After eliminating the fractions, the next step is to isolate the variable. This is typically done through basic algebraic operations.

Step 2: Isolate the Variable

To isolate x, subtract 3 from both sides of the equation:

8x - 3 3 10 3

8x 13

Now, divide both sides by 8:

x frac{13}{8}

Step 3: Check Your Solution

It's important to verify your solution by substituting it back into the original equation. This ensures the solution is correct.

Step 3: Check Your Solution

Substituting x frac{13}{8} back into the original equation:

frac{2}{3} left( frac{13}{8} right) - frac{1}{4} frac{5}{6}

Calculating the left side:

frac{2}{3} times frac{13}{8} frac{26}{24} frac{13}{12}

frac{1}{4} frac{3}{12}

Now add them together:

frac{13}{12} - frac{3}{12} frac{10}{12} frac{5}{6}

Since both sides are equal, the solution x frac{13}{8} is verified.

Solving Linear Equations Involving Decimals

Solving linear equations with decimals can be approached in a similar manner, but with a few additional considerations.

Method 1: Converting Decimals to Fractions

If the decimal can be written as a fraction, you can use the same methods for solving equations with fractions. Multiply both sides of the equation by the denominator to eliminate the decimals.

Method 2: Eliminating Decimals by Multiplication

Another approach is to multiply the entire equation by the appropriate multiple to eliminate the decimals. For example, if the equation contains decimals with denominators of 10, 100, or 1000, multiply the entire equation by the appropriate multiple to eliminate the decimals part of the equation.

Checking Your Solution

After solving the equation, it's important to check your solution by substituting it back into the original equation. This ensures the solution is correct.

Summary

Here's a summary of the steps involved in solving linear equations involving fractions and decimals:

Eliminate fractions by multiplying by the LCD.

Isolate the variable using basic algebraic operations.

Check your solution by substituting back into the original equation.

By following these steps, you can confidently solve any linear equation involving fractions and decimals.

Frequently Asked Questions (FAQs)

Q: Can I solve linear equations with decimals the same way as with fractions?

A: Yes, you can use similar methods to solve linear equations with decimals. Convert decimals to fractions or multiply through the equation to eliminate the decimals.

Q: What is the least common denominator (LCD)?

A: The least common denominator is the smallest number that is a multiple of all the denominators in a set of fractions.

Q: How do I check my solution?

A: Substitute the solved value back into the original equation to verify the solution is correct.