Overview
This lecture covers strategies for solving equations with fractions, focusing on techniques for clearing fractions and solving for the variable efficiently.
Solving Equations with One Fraction
- To eliminate a denominator, multiply both sides by its value (e.g., multiply both sides by 3 for an equation with x/3).
- After clearing the fraction, solve for x by isolating the variable using basic algebra.
- Alternatively, you can multiply both sides by the reciprocal of the fraction to solve more directly.
- When multiplying by the reciprocal, you can multiply then divide, or divide then multiply; dividing first is usually simpler with large numbers.
Solving Equations with Fractions and Constants
- If the equation includes a constant and a fraction, isolate the fractional term by adding or subtracting the constant from both sides first.
- Multiply both sides by the denominator to clear the fraction.
- Solve for x by dividing both sides by the remaining coefficient.
Solving Equations with Multiple Fractions
- Multiply both sides by the least common denominator (LCD) to eliminate all fractions from the equation.
- Distribute the LCD across all terms, simplifying each as you go.
- Once fractions are cleared, isolate x using basic algebraic steps.
Example: Handling Fractions with Different Denominators
- For equations with denominators like 2 and 4, the LCD is 4; multiply both sides by 4 to clear fractions efficiently.
- After clearing fractions, simplify and solve for x step-by-step: isolate x terms, combine constants, and divide to solve.
Key Terms & Definitions
- Reciprocal — The flipped version of a fraction, used to multiply and eliminate denominators (e.g., reciprocal of 2/3 is 3/2).
- Least Common Denominator (LCD) — The smallest number that is a multiple of all denominators in the equation, used to clear fractions.
- Isolate the variable — Get the variable alone on one side of the equation to solve for its value.
Action Items / Next Steps
- Practice solving equations with fractions using these techniques.
- Review homework problems involving equations with one or more fractional terms.