Linear Equation Solver

What are Linear Equations?

A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable. Linear equations have the general form:

ax + b = cx + d

where a, b, c, and d are constants, and x is the variable.

Characteristics

• The variable has a power of 1 (no x², x³, etc.)
• No variables in denominators
• No variables under radicals
• Graph as a straight line
• Has exactly one solution (unless inconsistent or identity)

Standard Steps to Solve

1. Simplify both sides (distribute, combine like terms)
2. Move all terms with variables to one side
3. Move all constant terms to the other side
4. Divide both sides by the coefficient of the variable
5. Check the solution by substitution

Solution Methods

Addition/Subtraction Property

You can add or subtract the same value from both sides of an equation without changing the solution.

If a = b, then a + c = b + c

Multiplication/Division Property

You can multiply or divide both sides by the same non-zero value without changing the solution.

If a = b and c ≠ 0, then ac = bc and a/c = b/c

Example Solution

Solve: 3x + 5 = 2x + 11

1. 3x + 5 = 2x + 11 (original equation)
2. 3x - 2x + 5 = 2x - 2x + 11 (subtract 2x from both sides)
3. x + 5 = 11 (simplify)
4. x + 5 - 5 = 11 - 5 (subtract 5 from both sides)
5. x = 6 (solution)

Types of Solutions

• One Solution: Most linear equations (e.g., x = 3)
• No Solution: Contradictory equations (e.g., 0 = 5)
• Infinite Solutions: Identity equations (e.g., 0 = 0)

Real-World Problems

• Distance and rate problems
• Age problems
• Money and coin problems
• Mixture problems
• Geometry applications

Science & Engineering

• Physics formulas
• Chemical equations
• Engineering calculations
• Economic models
• Statistical analysis

Advanced Mathematics

• Systems of equations
• Matrix operations
• Linear programming
• Differential equations
• Vector spaces