Elimination Method Calculator

The elimination method is a technique for solving a system of linear equations by adding or subtracting the equations to eliminate one variable. Once one variable is eliminated, you solve for the remaining variable and substitute back to find the other. It is one of the most widely used algebraic methods alongside substitution and matrix methods.

You enter the coefficients (a₁, b₁, c₁) for Equation 1 and (a₂, b₂, c₂) for Equation 2 in the form a·x + b·y = c. The calculator computes the determinant D = a₁·b₂ − a₂·b₁. If D ≠ 0, it applies Cramer's Rule — the same result as the elimination steps — to find unique values for x and y.

First, multiply one or both equations so that the coefficient of one variable is equal and opposite in both. Then add the equations to eliminate that variable and solve for the other. Finally, substitute the known value back into either original equation to find the eliminated variable. For example, to solve 2x + 3y = 8 and 3x + 2y = 7, multiply the first by 3 and the second by 2, then subtract to get y, and back-substitute to find x.

A system has no solution (inconsistent) when the two equations represent parallel lines — their determinant D = 0, but the constants are not proportional. In this case, elimination produces a contradiction like 0 = 5, meaning no values of x and y satisfy both equations simultaneously.

Infinite solutions occur when both equations are actually the same line (dependent system). The determinant D = 0 and the ratios a₁/a₂, b₁/b₂, and c₁/c₂ are all equal. Every point on that line is a valid solution, so there are infinitely many pairs (x, y) that satisfy the system.

The determinant D = a₁·b₂ − a₂·b₁ is the key value that determines the nature of the solution. If D ≠ 0, there is exactly one unique solution. If D = 0, the system is either inconsistent (no solution) or dependent (infinite solutions), depending on whether the constants are proportional.

Yes. The elimination method calculator handles any real number as a coefficient or constant, including decimals, fractions (entered as decimals), zero, and negative values. Just enter the numbers directly into the coefficient fields.

Both methods solve systems of linear equations, but they differ in approach. The substitution method isolates one variable in one equation and substitutes it into the other. The elimination method adds or subtracts multiples of equations to cancel out one variable entirely. Elimination is often faster when coefficients are already equal or can be made equal with simple multiplication.