Overview
This lecture covers strategies for solving two-step and multi-step algebraic equations, including those with fractions, parentheses, variables on both sides, and decimals.
Solving Basic Two-Step Equations
- To solve 3x + 5 = 17, subtract 5 from both sides, then divide by 3; x = 4.
- For 4x + 3 = 19, subtract 3, then divide by 4; x = 4.
- In 17 - 5x = 2, subtract 17, then divide by -5; x = 3.
Equations with Fractions
- For 9 = 3 + x/4, subtract 3, then multiply both sides by 4; x = 24.
- For 8 + x/3 = 12, subtract 8, then multiply by 3; x = 12.
Variables on Both Sides
- Move all x terms to one side and numbers to the other.
- Example: For 8 + 3x = 2 + 3x, subtract 2, then divide by 3; x = 4.
- For 13 - 2x = 4x, add 2x to both sides, then divide by 6; x = 3.
Combining Like Terms
- Combine all x terms on one side before isolating x.
- Example: 3x + 8 + 5x = 32, combine 3x and 5x, then solve; x = 3.
- For 7x + 2 - 3x = 26, combine like terms, subtract, then divide; x = 6.
Distributive Property (Parentheses)
- Distribute multiplication over terms inside parentheses before solving.
- Example: 3*(2x-4)+1 = 7; distribute, combine terms, isolate x; x = 3.
- For multiple parentheses: Use distributive property on both sides, then solve.
Multi-Fraction Equations
- Multiply both sides by the least common multiple (LCM) of denominators to clear fractions.
- Example: For fractions with denominators 3 and 2, multiply by 6 to eliminate fractions, then solve for x.
Decimals in Equations
- Multiply both sides by powers of 10 to eliminate decimals.
- Example: For 2x + 0.3 = 1.5, multiply by 10, then solve; x = 6.
Key Terms & Definitions
- Two-Step Equation — An equation solvable in two operations (e.g., add/subtract, then multiply/divide).
- Distributive Property — a(b + c) = ab + ac; used to remove parentheses.
- Combine Like Terms — Add or subtract variables and constants separately on the same side.
- Least Common Multiple (LCM) — The smallest number that is a multiple of two or more denominators; used to clear fractions.
Action Items / Next Steps
- Practice solving two-step and multi-step equations with variables, fractions, parentheses, and decimals.
- Review the distributive property and combining like terms.
- Complete homework assignments on solving equations as provided by your instructor.
- Read textbook sections on solving equations and the distributive property.