Linear Equation Solver - Solve Systems of Equations Step-by-Step
Equation Type
ax + b = 0
Enter coefficients for your equations. The solver uses Cramer's Rule for systems of equations.
Solution
Enter coefficients and click Solve
Understanding Linear Equations
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. Linear equations produce straight lines when graphed.
One variable: ax + b = 0
Two variables: ax + by = c
Three variables: ax + by + cz = d
Solution Types
- ✓Unique Solution: Equations intersect at exactly one point
- ∞Infinite Solutions: Equations represent the same line
- ✗No Solution: Equations are parallel and never intersect
Cramer's Rule
Cramer's Rule uses determinants to solve systems of linear equations efficiently. For a 2x2 system:
x = Dx / D
y = Dy / D
Where D is the main determinant and Dx, Dy are column substitutions
Real-World Applications
- Economics: Supply and demand equilibrium, cost analysis
- Engineering: Circuit analysis, structural load calculations
- Physics: Motion problems, force balance equations
- Chemistry: Chemical reaction balancing, mixture problems
- Business: Break-even analysis, resource allocation
Solving Tips
- • Always verify your solution by substituting back into the original equations
- • For 2 equations, check if lines are parallel (no solution) or identical (infinite solutions)
- • Cramer's Rule works best for small systems (2×2, 3×3)
- • If determinant = 0, the system either has no solution or infinite solutions
- • Graphing helps visualize the solution for 1 and 2-variable systems