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Solve 2x + 3y = 7 and x - y = 1 using Gaussian elimination
Key values: 2×2 system · Unique solution · x=2, y=1
Solve a 3-variable system commonly found in circuit analysis
Key values: 3×3 system · Cramer's Rule · 3 unknowns
Explore what happens when two equations have no intersection
Key values: 2×2 system · Inconsistent · No solution
Enter your system of linear equations to find the solution. Supports 2×2 and 3×3 systems with step-by-step Gaussian elimination, Cramer's rule, and matrix inverse.
A system of two linear equations in two unknowns represents two lines in the plane. Solving the system means finding where the lines intersect:
Unique solution
Lines intersect at exactly one point. The lines have different slopes.
No solution
Lines are parallel (same slope, different intercepts). The system is inconsistent.
Infinite solutions
Lines are identical (same equation). The system is dependent.
The coefficient determinant tells you which case you have before solving:
When , the solution is:
This is Cramer's rule for 2×2 systems.
Solve: and .
Step 1: Compute the determinant:
Step 2: Apply Cramer's rule:
Verify: ✓
A 2x2 system consists of two linear equations with two unknowns (typically and ). Geometrically, each equation represents a line in the plane, and solving the system means finding where the two lines intersect.
A 2x2 system has exactly one solution if the lines intersect (different slopes), no solution if the lines are parallel (same slope, different intercepts), or infinitely many solutions if the lines are identical (same equation).
The determinant of the coefficient matrix determines the outcome. If , there is a unique solution. If , the system has either no solution or infinitely many solutions, depending on whether the equations are contradictory or identical.
For a 2x2 system, Cramer's rule is fast: compute the determinant , then and . Alternatively, use elimination by adding or subtracting multiples of one equation from the other.
Common applications include break-even analysis (revenue = cost), mixture problems (combining solutions of different concentrations), supply and demand equilibrium, and simple circuit analysis using Kirchhoff's laws.