Solving Linear Equations: One-variable Equations

Definition: A linear equation in one variable is an equation that can be expressed in the form ax + b = 0, where:
- a and b are real numbers
- x is the variable
Basic Steps to Solve:
1. Isolate the variable:
  - Use inverse operations to get x by itself on one side of the equation.
2. Perform operations on both sides:
  - Whatever you do to one side of the equation, do the same to the other side.
3. Simplify:
  - Combine like terms and simplify both sides if possible.
Example: Solve for x in the equation 2x + 3 = 7
1. Subtract 3 from both sides: 2x = 4
2. Divide both sides by 2: x = 2
Types of Solutions:
- One solution: The equation can be solved to find a unique value for x.
- No solution: The equation simplifies to a false statement (e.g., 0 = 5).
- Infinite solutions: The equation simplifies to a true statement for all values of x (e.g., 0 = 0).
Special Cases:
- Identity: Equations that are true for all values of x (like 2(x + 1) = 2x + 2).
- Contradiction: Equations that have no solution (like 3x + 1 = 3x + 2).
Common Techniques:
- Combining like terms: Grouping similar terms to simplify.
- Using the distributive property: To eliminate parentheses (e.g., a(b + c) = ab + ac).
- Checking solutions: Substitute back into the original equation to verify correctness.
Applications:
- Used in various fields including physics, economics, and engineering for solving problems involving direct relationships.
Graphical Interpretation:
- A one-variable linear equation can be represented on a number line, and the solution is the point at which the equation holds true.
Practice Problems:
- Solve for x: 4x - 5 = 15
- Solve for x: 3(x - 2) + 4 = 10
- Determine the nature of the solution: 7x + 2 = 7x - 3
Key Formulas:
- General form: ax + b = 0
- Solution: x = -b/a (if a ≠ 0)

Definition of Linear Equations

A linear equation in one variable can be expressed as ax + b = 0, where a and b are real numbers and x is the variable.

Steps to Solve Linear Equations

Isolate the variable: Utilize inverse operations to solve for x.
Perform operations on both sides: Ensure both sides of the equation remain equal by performing the same operations.
Simplify: Combine like terms for clearer equations.

Example Solution

To solve 2x + 3 = 7:
- Subtract 3: 2x = 4
- Divide by 2: x = 2

Types of Solutions

One solution: Unique value for x found.
No solution: Results in a false statement (e.g., 0 = 5).
Infinite solutions: Results in a true statement for all x (e.g., 0 = 0).

Special Cases

Identity: Equations valid for all x (e.g., 2(x + 1) = 2x + 2).
Contradiction: Equations with no solutions (e.g., 3x + 1 = 3x + 2).

Common Techniques

Combining like terms: Simplifies equations by grouping similar terms.
Distributive property: Eliminates parentheses (e.g., a(b + c) = ab + ac).
Checking solutions: Substitute values back into the original equation to confirm correctness.