About this calculator
A system of linear equations in two variables contains two equations and two unknowns, in the form: a₁x+b₁y=c₁, a₂x+b₂y=c₂. Solving a system of equations means finding the values of x and y that satisfy both equations. Commonly used solution methods include substitution method, addition, subtraction and elimination method and Cramer's rule. Our free online quadratic equations solver uses Cramer's rule to provide a simple, fast, and accurate solution.
Cramer's rule uses determinants to solve a system of equations. Define the coefficient determinant D=a₁b₂-a₂b₁, the determinant of x Dx=c₁b₂-c₂b₁, and the determinant of y Dy=a₁c₂-a₂c₁. When D≠0, the system of equations has a unique solution: x=Dx/D, y=Dy/D. When D=0, if Dx=Dy=0, the system of equations has infinite solutions; otherwise, there is no solution.
Using the quadratic system solver is very simple and intuitive. Just enter the coefficients of the two equations, click the solve button, and get the x and y values instantly. This tool is particularly suitable for students to learn linear algebra, complete mathematics homework, verify calculation results, etc.
What it calculates
The system of equations calculator solves for values that satisfy two or more equations at the same time, commonly linear systems.
Formula
A 2-variable linear system can be written as a1x + b1y = c1 and a2x + b2y = c2. Substitution, elimination, or matrices can solve it.
Inputs
- Coefficients in each equation.
- Constant terms.
- Number of variables and equations.
Example
| System | Method | Result |
|---|---|---|
| x + y = 5; x - y = 1 | Elimination | x = 3, y = 2 |
| 2x + y = 7; x + y = 4 | Subtract equations | x = 3, y = 1 |
| x + y = 2; 2x + 2y = 4 | Dependent equations | Infinitely many solutions |
How to interpret the result
A unique solution means graphs meet at one point. No solution means they do not meet. Infinitely many solutions mean the equations share the same constraint.
Common mistakes
- Too few equations may not determine a unique solution.
- Apply elimination to both sides.
- Parallel lines correspond to no solution.
How to use
Using the quadratic system solver is very simple. First, put the two equations into standard form: a₁x+b₁y=c₁, a₂x+b₂y=c₂. For example, 2x+3y=8 and x-y=1 are already standard forms.
Then, enter the coefficients a₁, b₁, and c₁ of the first equation. Enter the coefficients a₂, b₂, and c₂ of the second equation. For example, for 2x+3y=8, a₁=2, b₁=3, c₁=8. For x-y=1, a₂=1, b₂=-1, c₂=1. Click the "Solve" button.
The calculator will solve using Cramer's rule and immediately display the x and y values. For example, the solution to the above system of equations is x=1, y=2. If the system of equations has no solution or infinite solutions, a corresponding prompt will be displayed. Click the "Reset" button to clear all input and start a new solution.
Main features
This linear equation solver has the following features: Use Cramer's rule to solve; automatically determine the solution situation (unique solution, infinite solutions, no solution); simultaneously display the values of x and y; high-precision calculation (retaining 4 decimal places); automatically detect invalid input; the interface is simple and intuitive, easy to use; fast response speed, solution results are displayed immediately; completely free, no registration or download required; supports desktop and mobile device access; suitable for student learning and linear algebra practice.
Use cases
The quadratic system solver is very useful in several scenarios. When students learn linear algebra, systems of linear equations in two variables are basic knowledge. You can use the solver to verify your calculations and understand Cramer's rule. As you complete your math homework, you can quickly check if your answers are correct.
In practical applications, systems of linear equations in two variables are used to solve various problems. Chicken and rabbit in the same cage problem: There are 10 chickens and rabbits in the cage with a total of 28 legs. How many chickens and rabbits are there? Suppose there are x chickens and y rabbits, then x+y=10, 2x+4y=28, and the solution is x=6, y=4. Proportion problem: Mix two solutions, the first containing 10% salt and the second containing 20% salt. To prepare 100 grams of a solution containing 15% salt, find the number of grams of each of the two solutions. Suppose the first type of x is grams and the second type is y, then x+y=100, 0.1x+0.2y=15, the solution is x=50, y=50.
Price question: It cost 23 yuan to buy 2 pens and 3 books. It cost 14 yuan to buy 1 pen and 2 books. Find the unit price of the pens and books. Assume that the pen is x yuan and the book is y yuan, then 2x+3y=23, x+2y=14, and the solution is x=4, y=5. In economics, systems of linear equations of two variables are also used in problems such as supply and demand balance and cost analysis. Whether for learning, application or research, the linear equations solver is a useful tool.