System of Equations Solver
Solve a 2×2 linear system for x and y with the determinant (Cramer) steps used on paper, including no-solution and infinitely-many cases.
One solution: the lines intersect.
- Write the system: 2x + y = 8 and x − y = 1.
- D = (2)(−1) − (1)(1) = −3.
- x = 3, y = 2.
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How to Use the System of Equations Solver
Enter the six coefficients of two linear equations in x and y, then select Solve System. The tool reports a unique pair, no solution, or infinitely many solutions, with the determinant steps.
- Type a₁, b₁, c₁ for the first equation a₁x + b₁y = c₁.
- Type a₂, b₂, c₂ for the second equation.
- Read x and y, or the classification if the lines do not meet in one point.
The Equation Solver handles one linear equation in x only. A quadratic in one variable belongs on the Quadratic Formula Calculator. Two points and a slope belong on the Slope Calculator — this page is the intersection of two lines written as equations.
Cramer’s Rule and Elimination
Dₓ replaces the x-column with the constants: c₁b₂ − c₂b₁. Dᵧ replaces the y-column: a₁c₂ − a₂c₁. When D is not zero the unique solution is those two quotients.
The same arithmetic is elimination. For 2x + y = 8 and x − y = 1 the y terms are opposites, so adding gives 3x = 9 and x = 3. Substitute into the second equation: 3 − y = 1, so y = 2. Cramer’s D = (2)(−1) − (1)(1) = −3, Dₓ = (8)(−1) − (1)(1) = −9, x = (−9)/(−3) = 3.
System of Equations Examples
A unique intersection
2x + y = 8 and x − y = 1. D = −3 ≠ 0, so one solution: x = 3, y = 2. Both original equations check.
Another unique pair
3x + 2y = 16 and x + y = 6. D = 1, x = 4, y = 2.
Decimals
0.5x + y = 4 and x − y = 2. D = −1.5, x = 4, y = 2.
Parallel lines
x + y = 3 and x + y = 5. D = 0 and the constants disagree, so there is no solution.
The same line twice
2x + 2y = 4 and x + y = 2. The second equation is half the first. D = Dₓ = Dᵧ = 0, so every point on the line is a solution.
Unique, None, or Infinitely Many
Two distinct non-parallel lines meet in one point. Parallel distinct lines never meet. If one equation is a multiple of the other, they are the same line and every point on it works.
Graphically that is the Slope Calculator’s two lines. This page classifies the algebra without asking you to graph.
Common System-of-Equations Mistakes
Switching Dx and Dy
Dx uses the constants in the x-column. Dy uses them in the y-column. Mixing them swaps x and y.
Treating D = 0 as “no solution” every time
D = 0 is either parallel (none) or the same line (infinitely many). Check Dx and Dy.
Solving only for x
A 2×2 system needs both unknowns. Substitute back to get y.
Entering a quadratic
x² does not belong here. Use the Quadratic Formula Calculator for one equation in x.
Using the one-variable Equation Solver
That page solves for x alone. Two variables need two equations.
System of Equations FAQs
How do you solve 2x + y = 8 and x − y = 1?
Add to get 3x = 9, so x = 3. Then 3 − y = 1 and y = 2. Cramer’s rule gives the same pair.
What does a zero determinant mean?
The lines are parallel. If the constants also match, they are the same line (infinitely many). If not, there is no solution.
Can I enter decimals?
Yes. Finite decimals and negatives are allowed in all six boxes.
Does this solve 3×3 systems?
No. This page is two equations and two unknowns.
Is this the same as the Equation Solver?
No. That tool is one linear equation in x. This tool is a 2×2 linear system in x and y.
Are my numbers sent to a server?
No. The calculation runs locally in your browser.
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Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
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Disclaimer
This solver is for general educational use. It solves two linear equations in two unknowns. It is not a computer-algebra system and does not handle nonlinear or larger systems.