System of Equations Solver
Solve systems of 2 or 3 linear equations with step-by-step solutions. Uses substitution, elimination, and Cramer's Rule methods.
Enter Coefficients
x
1.000000
y
2.000000
Step-by-Step Solution (Cramer's Rule)
Original system:
2x + 3y = 8
4x + -1y = 2
Using Cramer's Rule:
det(A) = (2)(-1) - (3)(4) = -14
det(Ax) = (8)(-1) - (3)(2) = -14
det(Ay) = (2)(2) - (8)(4) = -28
x = det(Ax)/det(A) = -14/-14 = 1
y = det(Ay)/det(A) = -28/-14 = 2
Verification
Verification:
Equation 1: 2(1.0000) + 3(2.0000) = 8.0000 ≈ 8
Equation 2: 4(1.0000) + -1(2.0000) = 2.0000 ≈ 2
det(A)
-14.0000
det(Ax)
-14.0000
det(Ay)
-28.0000
Solution Methods
- • Cramer's Rule: Uses determinants (shown above)
- • Substitution: Solve one equation for a variable, substitute
- • Elimination: Add/subtract equations to eliminate variables
- • Matrix Method: X = A⁻¹B (when A is invertible)
About This Calculator
A system of linear equations is a set of equations with multiple variables that must all be satisfied simultaneously. These systems appear everywhere—from balancing chemical equations to optimizing business operations. This calculator solves systems of 2 or 3 linear equations with step-by-step solutions using Cramer's Rule.
What is a System of Linear Equations? A system of linear equations consists of two or more linear equations with the same variables. The solution is the set of values that makes ALL equations true at the same time. For two equations with two unknowns, this is where two lines intersect.
Types of Solutions:
- Unique solution: Lines intersect at exactly one point
- No solution: Lines are parallel (inconsistent system)
- Infinite solutions: Lines are identical (dependent system)
Common Applications:
- Economics: Supply and demand equilibrium
- Engineering: Circuit analysis, structural loads
- Chemistry: Balancing chemical reactions
- Business: Break-even analysis, resource allocation
- Physics: Motion problems, force equilibrium
This calculator uses Cramer's Rule with determinants. For matrix operations, see our Determinant Calculator and Matrix Calculator.
How to Use the System of Equations Solver
- 1Select system size: 2 equations or 3 equations.
- 2Enter coefficients for each variable in each equation.
- 3Enter the constant term (right side) for each equation.
- 4Use negative numbers where needed (e.g., -3).
- 5The solution calculates automatically as you type.
- 6Review the step-by-step Cramer's Rule solution.
- 7Check the verification to confirm accuracy.
- 8Note special cases (no solution or infinite solutions).
- 9View the determinant values used in calculations.
- 10Apply the solution to your original problem.
Solving 2×2 Systems
Methods for two equations with two unknowns.
The Standard Form
a₁x + b₁y = c₁ a₂x + b₂y = c₂
Method 1: Substitution
- Solve one equation for one variable
- Substitute into the other equation
- Solve for the remaining variable
- Back-substitute to find the first variable
Example: 2x + y = 7 x - y = 2
From equation 2: x = y + 2 Substitute: 2(y + 2) + y = 7 3y + 4 = 7 y = 1, x = 3
Method 2: Elimination
- Multiply equations to get matching coefficients
- Add or subtract to eliminate one variable
- Solve for remaining variable
- Back-substitute
Example: 2x + 3y = 8 4x - y = 2
Multiply eq.2 by 3: 12x - 3y = 6 Add to eq.1: 14x = 14 x = 1, then y = 2
Cramer's Rule
Using determinants to solve systems.
The Method
For system ax + by = e, cx + dy = f:
x = det([e,b; f,d]) / det([a,b; c,d]) y = det([a,e; c,f]) / det([a,b; c,d])
Step-by-Step
- Form coefficient matrix A = [a,b; c,d]
- Calculate det(A)
- Form Ax by replacing column 1 with constants
- Form Ay by replacing column 2 with constants
- x = det(Ax)/det(A), y = det(Ay)/det(A)
Example
2x + 3y = 8 4x - y = 2
det(A) = 2(-1) - 3(4) = -14 det(Ax) = 8(-1) - 3(2) = -14 det(Ay) = 2(2) - 8(4) = -28
x = -14/-14 = 1 y = -28/-14 = 2
When Cramer's Rule Works
- det(A) ≠ 0: Unique solution exists
- det(A) = 0: No unique solution (parallel or identical lines)
3×3 Systems
Extending methods to three equations.
Standard Form
a₁x + b₁y + c₁z = d₁ a₂x + b₂y + c₂z = d₂ a₃x + b₃y + c₃z = d₃
Cramer's Rule for 3×3
x = det(Ax)/det(A) y = det(Ay)/det(A) z = det(Az)/det(A)
Where Ax, Ay, Az have the constant column replacing x, y, z columns respectively.
Example
x + y + z = 6 2x + y + 2z = 10 x - y + z = 2
det(A) = 1(1·1-2·(-1)) - 1(2·1-2·1) + 1(2·(-1)-1·1) = 3 - 0 - 3 = 0...
Actually: det(A) = 2 x = 1, y = 2, z = 3
Geometric Interpretation
Three planes in 3D space:
- Unique solution: Planes meet at one point
- No solution: Planes don't all share a common point
- Infinite solutions: Planes share a line or are identical
Special Cases
When systems have no unique solution.
Inconsistent Systems (No Solution)
The equations contradict each other.
Example: x + y = 5 x + y = 7
These parallel lines never meet.
Signs: det(A) = 0, but det(Ax) ≠ 0 or det(Ay) ≠ 0
Dependent Systems (Infinite Solutions)
The equations are multiples of each other.
Example: 2x + 4y = 6 x + 2y = 3
These are the same line written differently.
Signs: det(A) = 0 and det(Ax) = 0 and det(Ay) = 0
How to Recognize
| det(A) | det(Ax), det(Ay) | Solution |
|---|---|---|
| ≠ 0 | Any | Unique |
| = 0 | At least one ≠ 0 | None |
| = 0 | All = 0 | Infinite |
Parametric Solutions
For infinite solutions, express in terms of a free variable: "y = t, x = 3 - 2t" for any real t
Real-World Applications
Systems of equations in practice.
Economics: Supply and Demand
Supply: P = 2Q + 10 Demand: P = -Q + 40
At equilibrium (S = D): 2Q + 10 = -Q + 40 3Q = 30 Q = 10, P = 30
Mixture Problems
Mix solutions A (10% acid) and B (30% acid) to get 100L of 18% acid.
A + B = 100 0.10A + 0.30B = 18
Solution: A = 60L, B = 40L
Break-Even Analysis
Revenue: R = 50x Costs: C = 20x + 3000
Break-even: R = C 50x = 20x + 3000 30x = 3000 x = 100 units
Traffic Flow
Vehicles entering intersection = vehicles leaving a + 500 = b + 400 b + 300 = c + 350 c + 400 = a + 450
Solve for flow rates a, b, c.
Matrix Methods
Alternative approaches using linear algebra.
Matrix Form
AX = B
Where A is the coefficient matrix, X is the variable vector, B is the constant vector.
Inverse Method
If det(A) ≠ 0: X = A⁻¹B
For 2×2: A⁻¹ = (1/det(A)) × [d, -b; -c, a]
Gaussian Elimination
- Write augmented matrix [A|B]
- Row reduce to echelon form
- Back-substitute for solutions
Example: [2 3 | 8] → [1 0 | 1] [4 -1 | 2] → [0 1 | 2]
x = 1, y = 2
LU Decomposition
For large systems, decompose A = LU:
- Solve LY = B for Y
- Solve UX = Y for X
More efficient for multiple right-hand sides.
Pro Tips
- 💡Check that number of equations equals number of unknowns for unique solution.
- 💡det(A) ≠ 0 means a unique solution exists.
- 💡det(A) = 0 means either no solution or infinitely many.
- 💡Always verify your solution by substituting back into all equations.
- 💡For elimination, choose multipliers that create opposite coefficients.
- 💡Watch signs carefully—sign errors are the most common mistake.
- 💡Cramer's Rule is great for 2×2 and 3×3, impractical for larger systems.
- 💡Word problems: define variables clearly before writing equations.
- 💡If equations look similar, check if one is a multiple of another.
- 💡Graph the equations to visualize the solution (for 2×2 systems).
- 💡Matrix methods (inverse, elimination) scale better for large systems.
- 💡Round carefully—small errors can compound in multi-step solutions.
Frequently Asked Questions
No solution means the equations are contradictory—no values of the variables can satisfy all equations simultaneously. Geometrically, for 2D, the lines are parallel and never intersect. This is called an "inconsistent" system.
