To do algebra with fractions, you apply the same principles as with numerical fractions but carefully handle the variables involved. Here's a clear step-by- step guide:
Simplifying Algebraic Fractions
- Factor numerator and denominator if possible.
- Cancel out common factors between numerator and denominator.
- Example: 5x15\frac{5x}{15}155x can be simplified by dividing numerator and denominator by 5, resulting in x3\frac{x}{3}3x
Adding and Subtracting Algebraic Fractions
- Find a common denominator (often the least common denominator, LCD).
- Rewrite each fraction as an equivalent fraction with the LCD.
- Add or subtract the numerators while keeping the common denominator.
- Simplify the resulting fraction if possible.
- Example: 32m+13m\frac{3}{2m}+\frac{1}{3m}2m3+3m1
- LCD is 6m6m6m
- Rewrite as 96m+26m=116m\frac{9}{6m}+\frac{2}{6m}=\frac{11}{6m}6m9+6m2=6m11
Multiplying and Dividing Algebraic Fractions
- Multiply numerators together and denominators together.
- For division, multiply by the reciprocal of the divisor fraction.
- Simplify the result if possible.
- Example: 6x5÷2x3=6x5×32x=18x10x=95\frac{6x}{5}\div \frac{2x}{3}=\frac{6x}{5}\times \frac{3}{2x}=\frac{18x}{10x}=\frac{9}{5}56x÷32x=56x×2x3=10x18x=59
Solving Equations with Algebraic Fractions
- Eliminate fractions by multiplying both sides of the equation by the common denominator.
- Solve the resulting equation.
- Check for values that make denominators zero (excluded values).
- Example: Solve 8x=2\frac{8}{x}=2x8=2
- Multiply both sides by xxx: 8=2x8=2x8=2x
- Divide both sides by 2: x=4x=4x=4
- Check: 84=2\frac{8}{4}=248=2 correct
Important Notes
- Algebraic fractions are undefined when the denominator equals zero.
- Always factor and simplify where possible.
- Use common denominators to combine fractions.
- Multiply by reciprocals when dividing fractions
This approach covers the basics of working with fractions in algebra including simplification, addition, subtraction, multiplication, division, and solving fractional equations.
