What is how to do algebra with fractions?

Algebra with Fractions: A Quick Guide

Working with fractions in algebra can seem daunting, but it boils down to understanding a few key concepts and applying them consistently. Here's a breakdown:

1. Simplifying Fractions:

Before you start any algebraic manipulation, always simplify fractions. Find the greatest common factor (GCF) of the numerator and denominator and divide both by it. This makes the numbers smaller and easier to work with.

2. Adding and Subtracting Fractions:

• Common Denominator: To add or subtract fractions, they must have a common denominator. Find the least common multiple (LCM) of the denominators. This is your common denominator.
• Equivalent Fractions: Convert each fraction to an equivalent fraction with the common denominator. Remember to multiply both the numerator and denominator by the same factor.
• Add/Subtract Numerators: Once you have a common denominator, add or subtract the numerators. The denominator stays the same.
• Simplify: Always simplify your answer to its lowest terms.

3. Multiplying Fractions:

• Multiply the numerators together.
• Multiply the denominators together.
• Simplify the resulting fraction.
• Cross-simplifying (optional): Before multiplying, you can simplify fractions across the multiplication sign. Look for common factors between the numerator of one fraction and the denominator of the other.

4. Dividing Fractions:

• Invert and Multiply: Dividing by a fraction is the same as multiplying by its reciprocal (inverse). Flip the second fraction (the one you're dividing by) and then multiply.

5. Solving Equations with Fractions:

• Clear the Fractions: The easiest way to solve equations with fractions is often to clear the fractions entirely.
  • Find the least common denominator (LCD) of all the fractions in the equation.
  • Multiply every term on both sides of the equation by the LCD. This will cancel out the denominators.
  • Solve the resulting equation, which should now be free of fractions.

6. Working with Algebraic Fractions:

Algebraic fractions are fractions where the numerator and/or denominator contain variables. The same rules apply as with numerical fractions.

• Simplifying: Factor both the numerator and denominator and cancel out any common factors.
• Adding, Subtracting, Multiplying, and Dividing: Follow the same rules as with numerical fractions, but remember to factor and simplify whenever possible.
• Solving Equations: Use the same techniques for solving equations with numerical fractions, being especially careful with factoring and distributing when clearing denominators.

Important Considerations:

• Excluded Values: Be mindful of values that would make the denominator of a fraction equal to zero. These values are excluded from the domain of the expression. For example, in the expression 1/(x-2), x cannot be 2.
• Factoring: Factoring is your best friend! It allows you to simplify fractions, find common denominators, and solve equations.
• Distributive Property: Don't forget to correctly use the distributive property when multiplying expressions involving fractions.