Systems of Equations Solver (2×2)
How It Works
This systems of equations solver (2×2) uses established formulas to provide accurate results.
The basic rule:
- a₁x + b₁y = c₁
- a₂x + b₂y = c₂
- x = (c₁b₂ - c₂b₁) / (a₁b₂ - a₂b₁)
- y = (a₁c₂ - a₂c₁) / (a₁b₂ - a₂b₁)
Results are estimates. Consult a professional for critical decisions.
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Explore Allday Everyday Math →Frequently Asked Questions
What is a system of equations?
A system of equations is a set of two or more equations with the same variables. The solution is the point (x, y) that satisfies all equations simultaneously — graphically, where the lines intersect.
When does a system have no solution?
A system has no solution when the lines are parallel — they have the same slope but different y-intercepts. The determinant a₁b₂ - a₂b₁ equals zero and the lines are not identical.
What is the elimination method?
In elimination, you multiply one or both equations by constants so that adding or subtracting them eliminates one variable. Then solve for the remaining variable and back-substitute.
What is Cramer's Rule?
Cramer's Rule uses determinants to solve a system. For a 2×2 system, x = (c₁b₂ - c₂b₁)/D and y = (a₁c₂ - a₂c₁)/D where D = a₁b₂ - a₂b₁.