Solution
- $x$: —
- $y$: —
Step-by-step
Explanation:
2x2 System of Equations Solver
This tool solves a system of two linear equations with two variables and shows the work step by step using Cramer’s rule. Enter the coefficients for $a_1x+b_1y=c_1,\quad a_2x+b_2y=c_2$ and the solver will compute the determinants $D$, $D_x$, and $D_y$ to determine whether the system has a unique solution, no solution, or infinitely many solutions. For more algebra tools, explore Algebra.
Solve a 2x2 system in the form:
$$a_1x + b_1y = c_1$$
$$a_2x + b_2y = c_2$$
What Is a 2x2 System of Equations?
A 2x2 system is two linear equations in the same two variables, usually $x$ and $y$. Geometrically, each equation is a line. The solution is the intersection point of the two lines. Depending on the lines, a system can have one intersection (unique solution), no intersection (no solution), or infinitely many intersections (the same line).
How This Solver Works
This solver uses Cramer’s rule. It computes three determinants: the main determinant $D$, plus $D_x$ and $D_y$ where the constants replace one column at a time. These values tell you both the solution type and, when it exists, the exact values of $x$ and $y$.
$$D=a_1b_2-a_2b_1,\quad D_x=c_1b_2-c_2b_1,\quad D_y=a_1c_2-a_2c_1$$
$$D\ne 0 \Rightarrow x=\frac{D_x}{D},\; y=\frac{D_y}{D}$$
How to Interpret the Determinants
Cramer’s rule gives a clean classification:
- Unique solution: if $D \ne 0$, the system has exactly one solution $(x,y)$.
- Infinitely many solutions: if $D=0$ and $D_x=0$ and $D_y=0$, the two equations represent the same line.
- No solution: if $D=0$ but at least one of $D_x$ or $D_y$ is nonzero, the lines are parallel and never intersect.
If you only need to solve a single equation in one variable, use the Linear Equation Solver. If your equation includes an $x^2$ term, use the Quadratic Solver.
Input Notes
Coefficients can be integers or decimals. The tool formats results using the configured decimal precision. If the determinants are extremely close to zero (because of rounding), the solver treats them as zero to keep the classification stable.
Worked Examples With Steps
Example 1: Unique solution
Solve a system with one intersection point.
Solve: $$\begin{cases} 2x + 3y = 13\\ x - y = 1 \end{cases}$$
Compute determinants: $$D=(2)(-1)-(1)(3)=-2-3=-5$$ $$D_x=(13)(-1)-(1)(3)=-13-3=-16$$ $$D_y=(2)(1)-(1)(13)=2-13=-11$$
Since $D\ne 0$: $$x=\frac{D_x}{D}=\frac{-16}{-5}=3.2,\quad y=\frac{D_y}{D}=\frac{-11}{-5}=2.2$$
Example 2: Infinitely many solutions
The equations are multiples of each other, so they represent the same line.
Solve: $$\begin{cases} x + 2y = 4\\ 2x + 4y = 8 \end{cases}$$
Compute determinants: $$D=(1)(4)-(2)(2)=4-4=0$$ $$D_x=(4)(4)-(8)(2)=16-16=0$$ $$D_y=(1)(8)-(2)(4)=8-8=0$$
Since $D=0$ and $D_x=0$ and $D_y=0$, the system has infinitely many solutions.
Example 3: No solution
The lines are parallel, so they never intersect.
Solve: $$\begin{cases} x + y = 2\\ 2x + 2y = 5 \end{cases}$$
Compute determinants: $$D=(1)(2)-(2)(1)=2-2=0$$ $$D_x=(2)(2)-(5)(1)=4-5=-1$$ $$D_y=(1)(5)-(2)(2)=5-4=1$$
Since $D=0$ but at least one of $D_x$ or $D_y$ is nonzero, the system has no solution.
Keep exploring Mathematics or jump back to Calculators to browse more tools.
Questions About the 2x2 System Solver
Quick answers about determinants, Cramer's rule, and solution types.
It solves two linear equations in two variables written as $a_1x+b_1y=c_1$ and $a_2x+b_2y=c_2$, and reports whether the system has a unique solution, no solution, or infinitely many solutions.
Cramer's rule solves a 2x2 system using determinants. Compute $D=a_1b_2-a_2b_1$, $D_x=c_1b_2-c_2b_1$, and $D_y=a_1c_2-a_2c_1$.
The system has a unique solution when $D\ne 0$. In that case, $x=\frac{D_x}{D}$ and $y=\frac{D_y}{D}$.
It has infinitely many solutions when $D=0$ and $D_x=0$ and $D_y=0$. This means both equations represent the same line.
It has no solution when $D=0$ but at least one of $D_x$ or $D_y$ is nonzero. This typically corresponds to parallel lines that never intersect.
Yes. The solver accepts decimal and negative coefficients for $a_1,b_1,c_1,a_2,b_2,c_2$, then formats results using the tool’s configured decimal precision.
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