Key Concepts:
- Linear Equation: An equation in the form (ax + b = c), where (a), (b), (c) are constants and (x) is the variable.
- Types of Solutions:
- One Solution: The equation simplifies to a single value of (x) (e.g., (x = 5)).
- No Solution: The equation simplifies to a false statement (e.g., 3 = 5).
- Infinitely Many Solutions: The equation simplifies to a true statement (e.g., (4x + 2 = 4x + 2)).
Skills Covered:
- Identifying different types of solutions by simplifying both sides of the equation.
Example Problems:
- Solve: (2x + 3 = 9)
- Subtract 3 from both sides: (2x = 6)
- Divide by 2: (x = 3) (One solution)
- Solve: (4x + 7 = 4x + 2)
- Subtract (4x) from both sides: 7 = 2
- This is false, so there is no solution.
- Solve: (x + 3 = x + 3)
- Identical sides make a true statement, so there are infinitely many solutions