Can I enter fractions?

Yes. Each cell accepts integers, decimals, negative values, and common fractions such as 1/2.

What does a zero main determinant mean?

It means the system does not have a single intersection point. The equations may represent the same line or parallel lines.

Is this Cramer's rule or substitution?

The solver uses Cramer's rule. Substitution is used only to check the final pair.

How do I check the answer?

Substitute the x and y values into both original equations. The calculator can show that check.

2x2 System of Equations Solver

Q: What if my equations are not in standard form?

Move x and y terms to the left and constants to the right before entering the coefficients.

Solve a 2x2 system of linear equations with Cramer's rule, determinants, solution classification, fractions, and substitution check.

Enter coefficients for two equations.

x, row 1

y, row 1

constant 1

x, row 2

y, row 2

constant 2

Virtual keypad for the active cell

Examples

Solution

Main determinant

-2

x determinant

-2

y determinant

-2

Method values

The calculator computed the main determinant and two replacement determinants. If the main determinant is nonzero, the solution is unique.

The check substitutes the values back into both equations.

A 2x2 linear system is solved from two coefficient rows. The calculator shows variable values, determinant checks, and substitution verification.

How to solve a 2x2 system by coefficients

This page is built for coefficient input. Each row is already in linear form, so you enter the x coefficient, the y coefficient, and the constant. The calculator applies Cramer's rule, shows the determinants, classifies special cases, and checks the ordered pair.

a_{11} x + a_{12} y = b_{1}

x and y are the unknowns; a11, a12, and b1 belong to the first equation.

a_{21} x + a_{22} y = b_{2}

a21, a22, and b2 belong to the second equation.

Column	What to enter
First column	coefficient of x
Second column	coefficient of y
Third column	constant on the right side

Enter the coefficients for both equations
Compute the main determinant
Compute the x and y replacement determinants
Divide by the main determinant when it is not zero
Check the values in the original equations

Cramer's rule for two unknowns

Cramer's rule builds the main determinant from the coefficient matrix. Each replacement determinant swaps one coefficient column with the constants. If the main determinant is nonzero, the ratios give the unique solution.

Δ = a_{11} a_{22} - a_{12} a_{21}

Delta is the main determinant of the coefficient matrix.

Δ_{x} = b_{1} a_{22} - a_{12} b_{2}

Delta x replaces the x column with the constants.

Δ_{y} = a_{11} b_{2} - b_{1} a_{21}

Delta y replaces the y column with the constants.

x = \frac{Δ _{x}}{Δ}, y = \frac{Δ _{y}}{Δ}

These ratios are valid when Delta is not zero.

Possible outcomes

Algebra test	Result	Graph meaning
Main determinant is nonzero	One solution	Lines intersect
All determinants are zero	Infinitely many solutions	Lines coincide
Main determinant is zero and a replacement determinant is nonzero	No solution	Lines are parallel

Preparing equations for coefficient input

If your system is written in textbook form, move variable terms to the left and constants to the right before entering the coefficients.

x = 2 y - 3

x and y appear before moving terms.

x - 2 y = - 3

x and y after moving terms into coefficient form.

Use this page when	Use free-form input when
The coefficients are already known	You need to paste full equations
You want a quick Cramer's rule calculation	You need parentheses expanded
The system is already linear	The expression still needs algebra cleanup

Coefficient workflow

This solver focuses on the 2x2 coefficient table. It does not parse full equation text.

Geometric meaning

Two lines on a plane

Each equation represents a line. One solution is the intersection point, infinitely many solutions means the same line, and no solution means parallel lines.

Frequently Asked Questions

Sources and References

System of Equations
Wolfram MathWorld

Calculations are based on the listed reference sources. Links open in a new tab.

Updated: June 2, 2026

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