System of Equations Calculator
Use this calculator to solve two linear equations in two unknowns for algebra, intersection points, and practical mixture or cost problems.
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Result
The formula
A 2x2 system can be written as a1x + b1y = c1 and a2x + b2y = c2. If the determinant D = a1b2 − a2b1 is non-zero, the system has one unique solution.
Worked example
- Given 2x + y = 11 and x − y = 1.
- Add the equations: 3x = 12.
- So x = 4, then y = 3.
Result: x = 4, y = 3
How the system is solved
The calculator uses determinant checks to determine whether the system has one solution, no solution, or infinitely many solutions.
Unique solution
When D ≠ 0, use Cramer’s rule or elimination to compute x and y.
No solution
If the coefficients are proportional but the constants are not, the lines are parallel and never intersect.
Infinite solutions
If both equations describe the same line, every point on that line is a solution.
Real-world interpretation
A solution is the intersection point of two lines. Systems appear in budgeting, mixing, break-even analysis, and balance problems.
Interesting facts
The determinant tells you the outcome
A non-zero determinant means exactly one intersection point; a zero determinant means the lines are parallel or identical.
Systems model intersections
Solving a system of equations is the same as finding where two lines meet.
Used in real-world calculations
Systems of equations are used in mixture problems, cost/profit intersections, resource allocation, and motion problems where two constraints meet.
Frequently asked questions
Then the system has either no solution or infinitely many solutions.
Yes. The solution is often not a whole number.
Yes. Negative coefficients and constants are valid.
Yes, this calculator is designed for two equations in two unknowns.
It is the point where the two lines intersect.