Overview
The lecture explains how to solve linear equations in one variable, outlines key solution steps, and discusses special types of equations with no or infinite solutions.
Definition of a Linear Equation in One Variable
- A linear equation in one variable can be written as ax + b = c, where a, b, and c are real numbers and a ≠ 0.
- Solving the equation means finding the value of the variable that makes the equation true.
Solving Linear Equations: Steps and Properties
- To solve, rewrite the equation using simpler, equivalent forms until it has the format variable = number.
- Equivalent equations have the same solutions.
- Use properties of equality:
- Add/subtract the same number on both sides to get an equivalent equation.
- Multiply/divide both sides by the same nonzero number to get an equivalent equation.
- The goal is to isolate the variable on one side of the equation.
Example Solutions and Techniques
- Collect all variable terms on one side and constants on the other.
- Combine like terms to simplify before isolating the variable.
- Use the distributive property to remove parentheses.
- Check your solution by substituting it back into the original equation.
Standard Steps for Solving Linear Equations
- Step 1: Clear fractions by multiplying both sides by the least common denominator (LCD).
- Step 2: Use distributive property to remove grouping symbols (parentheses).
- Step 3: Combine like terms on each side.
- Step 4: Use the addition/subtraction property to get variables on one side and numbers on the other.
- Step 5: Use the multiplication/division property to isolate the variable.
- Step 6: Check the solution in the original equation.
Special Types of Equations
- Contradiction: An equation with no solution (variables cancel and a false statement remains, e.g., 8 = 2).
- Identity: An equation true for all real numbers (variables cancel and a true statement remains, e.g., 0 = 0).
Key Terms & Definitions
- Linear equation — An equation of the form ax + b = c with one variable and a ≠ 0.
- Equivalent equations — Equations with exactly the same solutions.
- Property of equality — Rules allowing addition, subtraction, multiplication, or division of both sides by the same number.
- Contradiction — An equation with no possible solutions.
- Identity — An equation true for all real numbers for which it is defined.
- Least common denominator (LCD) — The smallest common multiple of all denominators in an equation.
Action Items / Next Steps
- Practice solving linear equations, including checking your solutions.
- Try identifying contradictions and identities in given equations.