Substitution Method Calculator

The substitution method is an algebraic technique where you solve one equation for one variable, then substitute that expression into the other equation. This reduces the system to a single equation with one unknown, which you can solve directly.

Enter your equations in the standard form ax + by = c. For Equation 1, provide the coefficients a₁, b₁ and constant c₁. For Equation 2, provide a₂, b₂ and c₂. For example, 2x + y = 8 would be entered as a=2, b=1, c=8.

A system has no solution (it is inconsistent) when the two equations represent parallel lines that never intersect. Algebraically, this occurs when the determinant (a₁·b₂ − a₂·b₁) equals zero but the equations are not multiples of each other.

A system has infinitely many solutions (it is dependent) when both equations represent the same line. This happens when one equation is a scalar multiple of the other, meaning the determinant is zero and the ratio of constants also matches.

After finding (x, y), the calculator substitutes those values back into both original equations and checks that each side balances. The verification results shown should both equal the respective constants c₁ and c₂.

Yes. You can enter any real-number coefficients, including negative values. For example, entering b₂ = −1 represents the term −y in your second equation.

The determinant D = a₁·b₂ − a₂·b₁ determines whether the system has a unique solution. If D ≠ 0, there is exactly one solution. If D = 0, the lines are either parallel (no solution) or identical (infinite solutions).

Absolutely. The calculator works with any real number inputs and returns decimal solutions rounded to four decimal places, so fractional and irrational-looking results are handled correctly.