# Two-Variable Linear Equation Solver

This calculator solves a system of equations of first degree with two unknowns. You enters six coefficients, it finds the values of the unknowns and detects cases of no solution or infinite solutions.

The system of equations of first degree with two unknowns is:
$\left\{{ax+by=c\atop dx+ey=f$
where a, b, c, d, e and f are real numbers, and x and y are unknowns.

You should enter the coefficients a, b, c, d, e and f in the form below, and the calculator displays x and y.

The formulas used to solve the equations can be found below the calculator.

#### Solving a system of equations of first degree with two uknknowns

Digits after the decimal point: 2
x

y

### Solving the equations

To solve the equations you should find the values of x and y that satisfy both equations simultaneously.

The steps involved in solving a system of equations of first degree with two unknowns include eliminating one of the unknowns by adding or subtracting the two equations and then solving for the remaining unknown. This can be done using either substitution or elimination methods. Substitution involves solving one equation for one variable and substituting that value into the other equation, while elimination involves adding or subtracting the two equations to eliminate one of the variables.

However, there are common formulas:
$x=\frac{ec-bf}{ae-db}\\y=\frac{af-dc}{ae-db}$
These formulas are easy to remember, if you introduce the concept of determinant of the second order as
$\left|\begin{matrix} p & q \\ r & s \end{matrix} \right| = ps-rq$
Then the solution of the equations can be written as
$x=\frac{\left|\begin{matrix} c & b \\ f & e \end{matrix} \right|}{\left|\begin{matrix} a & b \\ d & e \end{matrix} \right|}\\y=\frac{\left|\begin{matrix} a & c \\ d & f \end{matrix} \right|}{\left|\begin{matrix} a & b \\ d & e \end{matrix} \right|}$
ie each of the unknowns is equal to the fraction, the denominator of which is the determinant consisting of the coefficients of the unknowns and the numerator is obtained from this determinant by replacement of coefficients of the corresponding unknown to the absolute term.

There are three different solitions possible:

1. Coefficients at unknowns in equations are disproportional
$\frac{a}{d}<>\frac{b}{e}$
in this case the system of equations has a single solution, presented by formula

2. Coefficients at unknowns are proportional, but disproportionate to free terms
$\frac{a}{d}=\frac{b}{e}<>\frac{c}{f}$
in this case the system of equations has no solutions, because we have here contradictory equations.

3. All coefficients of equations are proportional
$\frac{a}{d}=\frac{b}{e}=\frac{c}{f}$
The system of equations has an infinite set of solutions, because we have actually one equation instead of two.
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