How the Substitution Method Works
The substitution method solves a system of two linear equations in two unknowns, written in standard form as a₁x + b₁y = c₁ and a₂x + b₂y = c₂. The idea is to reduce two unknowns to one: solve one equation for a single variable, substitute that expression into the other equation, solve the resulting one-variable equation, and then substitute that value back in to find the second variable.
Formula and method
Starting from equation 1, isolate y (assuming b₁ ≠ 0): y = (c₁ − a₁x) / b₁. Substituting this into equation 2 and solving for x gives x = (c₁b₂ − c₂b₁) / (a₁b₂ − a₂b₁). Once x is known, y follows from y = (a₁c₂ − a₂c₁) / (a₁b₂ − a₂b₁). Both formulas share the same denominator, D = a₁b₂ − a₂b₁, called the determinant of the system. This calculator performs exactly that elimination-by-substitution for you and reports x, y, D, and the solution point.
Interpreting the determinant
If D ≠ 0, the two equations represent two lines that cross at exactly one point, so the system has a unique solution. If D = 0, the lines are parallel: check whether the constants are also proportional in that same ratio. If they are, the two equations describe the same line, so every point on it is a solution (infinitely many solutions). If they are not, the lines are parallel but distinct and never meet, so the system has no solution.
Common sources of error
- Wrong sign on a coefficient: moving a term across the equals sign flips its sign — double-check that each equation is truly in a₁x + b₁y = c₁ form before entering coefficients.
- Substituting into the same equation: the isolated expression from equation 1 must be substituted into equation 2, not back into equation 1, or you will get an identity instead of a solution.
- Dropping a negative during distribution: when the substituted expression is multiplied through, distribute the sign carefully across every term.
Checking your result
Verify a solution by plugging x and y back into both original equations — each side should balance exactly. This calculator performs that check automatically and reports it in the results description.