Overview
This lecture covers solving linear algebraic equations, including collecting like terms, expansion and simplification, working with fractions, and solving word problems using systems of equations.
Solving Linear Equations: Collecting Like Terms
- To solve equations, move variables (x terms) to one side and constants to the other.
- Example: For 7 - 3x = 2x - 3, rearrange to group x terms and constants, giving -5x = -10, so x = 2.
Expanding Brackets and Simplifying
- Expand brackets before collecting like terms in equations involving parentheses.
- Distribute negative signs carefully when expanding.
- Example: 10 - 4(2x - 1) = -2(3 - x) expands to 14 - 8x = -6 + 2x, which simplifies to -10x = -20, so x = 2.
Working with Fractions in Equations
- To clear fractions, multiply both sides by the denominator or use cross-multiplication.
- Example: (3x - 1)/2 = 4 becomes 3x - 1 = 8. Solve to get x = 3.
Solving Word Problems Using Equations
- Assign variables to unknowns and translate word statements into equations.
- Example: "Po is 5 years older than Kyo. Their ages sum to 25." Let Po = x, Kyo = y, so x = y + 5 and x + y = 25.
- Substitute to form an equation with one variable: (y + 5) + y = 25 ⇒ 2y = 20 ⇒ y = 10.
- Substitute back: x = 10 + 5 = 15.
Key Terms & Definitions
- Like Terms — Terms with the same variable parts, which can be combined through addition or subtraction.
- Expansion — Multiplying out expressions within brackets to remove parentheses.
- Transpose — Moving a term across the equals sign, changing its sign.
- Substitution — Replacing a variable with its equivalent expression or value.
Action Items / Next Steps
- Practice more exam-style algebra problems, focusing on expansion, collecting like terms, and solving equations.
- Review the method for solving word problems by translating them into algebraic equations.