Linear Equation Solver

Linear Equation Solver - Step-by-Step Solutions

📐 Solve linear equations with detailed step-by-step solutions. Supports simple equations, complex multi-variable equations, and systems of two equations with graph visualization.

What is a Linear Equation?

A linear equation is an algebraic equation where each term is either a constant or the product of a constant and a single variable. The graph of a linear equation is always a straight line.

Types of Linear Equations

1. Simple Form: ax + b = c

• Example: 2x + 3 = 11
• Solution: x = 4

2. Standard Form: ax + b = cx + d

• Example: 3x + 5 = 2x + 8
• Solution: x = 3

3. System of Equations:

• Two or more equations with multiple variables
• Example: 2x + 3y = 8 and x - y = 1
• Solution: x = 2, y = 1.33

Solving Simple Equations (ax + b = c)

Steps:

1. Subtract b from both sides: ax = c - b
2. Divide both sides by a: x = (c - b) / a

Example: 2x + 3 = 11

Step 1: Subtract 3 from both sides
    2x + 3 - 3 = 11 - 3
    2x = 8

Step 2: Divide both sides by 2
    2x / 2 = 8 / 2
    x = 4

Verification: 2(4) + 3 = 8 + 3 = 11 ✓
            

Solving Standard Form (ax + b = cx + d)

Steps:

1. Move all x terms to one side: ax - cx = d - b
2. Factor out x: (a - c)x = d - b
3. Divide by coefficient: x = (d - b) / (a - c)

Example: 3x + 5 = 2x + 8

Step 1: Subtract 2x from both sides
    3x - 2x + 5 = 2x - 2x + 8
    x + 5 = 8

Step 2: Subtract 5 from both sides
    x + 5 - 5 = 8 - 5
    x = 3

Verification: 3(3) + 5 = 9 + 5 = 14
              2(3) + 8 = 6 + 8 = 14 ✓
            

System of Equations - Substitution Method

Example:

Equation 1: 2x + 3y = 8
Equation 2: x - y = 1

Step 1: Solve Equation 2 for x
    x = y + 1

Step 2: Substitute into Equation 1
    2(y + 1) + 3y = 8
    2y + 2 + 3y = 8
    5y + 2 = 8
    5y = 6
    y = 1.2

Step 3: Find x
    x = y + 1 = 1.2 + 1 = 2.2

Solution: x = 2.2, y = 1.2
            

System of Equations - Elimination Method

Example:

Equation 1: 2x + 3y = 8
Equation 2: x - y = 1

Step 1: Multiply Equation 2 by 2
    2x - 2y = 2

Step 2: Subtract from Equation 1
    (2x + 3y) - (2x - 2y) = 8 - 2
    2x + 3y - 2x + 2y = 6
    5y = 6
    y = 1.2

Step 3: Substitute back
    x - 1.2 = 1
    x = 2.2
            

Special Cases

No Solution (Parallel Lines):

• Example: 2x + 3 = 2x + 5
• Result: 0x = 2 (impossible)
• Lines have same slope, different y-intercepts

Infinite Solutions (Same Line):

• Example: 2x + 4 = 2x + 4
• Result: 0x = 0 (always true)
• Equations represent the same line

Graphing Linear Equations

Slope-Intercept Form: y = mx + b

• m = slope (rise over run)
• b = y-intercept (where line crosses y-axis)

Standard Form: Ax + By = C

• Find x-intercept: set y = 0
• Find y-intercept: set x = 0
• Plot both points and draw line

Applications

• Physics: Speed, distance, time problems
• Economics: Supply and demand curves
• Chemistry: Concentration calculations
• Engineering: Force and motion analysis
• Business: Break-even analysis

Common Mistakes

• Sign errors: Forgetting to change signs when moving terms
• Division by zero: Check coefficient before dividing
• Order of operations: Solve parentheses first
• Fractions: Find common denominator before solving

💡 Pro Tip: Always verify your solution by substituting it back into the original equation! This catches arithmetic errors and confirms the answer. For systems of equations, check both equations. Also, when dealing with fractions, consider multiplying both sides by the LCD (Least Common Denominator) at the start to eliminate fractions entirely - it makes the algebra much cleaner!