Elimination Method Calculator

Enter Calculation

a₁ (x coefficient, eq 1)

b₁ (y coefficient, eq 1)

c₁ (constant, eq 1)

a₂ (x coefficient, eq 2)

b₂ (y coefficient, eq 2)

c₂ (constant, eq 2)

Results

Solution of the System

x = 3, y = 2

unique solution (lines intersect)

x	3
y	2
Determinant (a₁b₂ − a₂b₁)	-5

What Is the Elimination Method?

The elimination method (also called the addition method) is a technique for solving a system of two linear equations in two unknowns. You scale the equations so that one variable cancels when the equations are added or subtracted, then back-substitute. This calculator does the algebra for you: enter the six coefficients of the system \(a_1 x + b_1 y = c_1\) and \(a_2 x + b_2 y = c_2\), and it returns the exact values of x and y.

A 2x2 linear system corresponds to two lines whose intersection is the solution (x, y).

How to Use the Calculator

Type the coefficients from each equation into the matching boxes. The first row is equation 1 (\(a_1\), \(b_1\), \(c_1\)) and the second row is equation 2 (\(a_2\), \(b_2\), \(c_2\)). Coefficients may be negative or decimal. Press calculate to see the solution along with the determinant \(a_1 b_2 - a_2 b_1\), which tells you whether a unique solution exists.

The Formula Explained

Using Cramer's rule (equivalent to elimination), the solution is $$x = \frac{c_1 b_2 - c_2 b_1}{a_1 b_2 - a_2 b_1}, \qquad y = \frac{a_1 c_2 - a_2 c_1}{a_1 b_2 - a_2 b_1}$$ The shared denominator \(a_1 b_2 - a_2 b_1\) is the coefficient determinant. If it is zero, the lines are parallel — either no solution (inconsistent) or infinitely many (the same line).

Scaling the equations so one variable's coefficients match, then subtracting, eliminates that variable.

Worked Example

Solve \(2x + 3y = 12\) and \(x - y = 1\). Here \(a_1=2\), \(b_1=3\), \(c_1=12\), \(a_2=1\), \(b_2=-1\), \(c_2=1\). The determinant is $$2(-1) - 1(3) = -5$$ Then $$x = \frac{12\cdot-1 - 1\cdot3}{-5} = \frac{-15}{-5} = 3, \qquad y = \frac{2\cdot1 - 1\cdot12}{-5} = \frac{-10}{-5} = 2$$ So \(x = 3\), \(y = 2\).

FAQ

What if the determinant is zero? The system has no unique solution. The calculator checks consistency and reports either "no solution" (parallel lines) or "infinitely many solutions" (identical lines).

Can I enter decimals or fractions? Decimals work directly. Convert a fraction to its decimal value first (e.g. \(1/2 \to 0.5\)).

Is this the same as substitution? Both methods give the same answer; elimination cancels a variable by combining equations, while substitution isolates one variable first.