Solve two linear equations and verify their unique intersection using substitution-style equation checks.
Complete the required fields and select the calculate button.
Parallel or identical equations do not have one unique intersection. The calculator identifies those systems as invalid for a single-point result.
This calculator solves two linear equations in two unknowns. Enter each equation in standard form, a₁x + b₁y = c₁ and a₂x + b₂y = c₂. The result is the coordinate pair where the two lines meet.
Although the page is named for substitution, the numerical shortcut uses determinants, which gives the same unique solution. The result should always be checked by replacing x and y in both original equations.
The calculator rejects a zero determinant because parallel or identical lines do not have one unique ordered pair.
D = a₁b₂ − a₂b₁; x = (c₁b₂ − c₂b₁) ÷ D; y = (a₁c₂ − a₂c₁) ÷ D
These formulas apply when D ≠ 0. In manual substitution, isolate one variable in either equation, insert that expression into the other equation, solve, and back-substitute. Both paths produce the same pair.
| Symbol | Meaning |
|---|---|
| a₁, b₁ | Coefficients of x and y in the first equation. |
| c₁ | Constant on the first equation’s right side. |
| a₂, b₂, c₂ | Corresponding values from the second equation. |
| D | Determinant; nonzero means one unique solution. |
| Precision | Decimal places shown in x and y. |
For x + y = 5 and x − y = 1, enter a₁ = 1, b₁ = 1, c₁ = 5, a₂ = 1, b₂ = −1, and c₂ = 1. Then D = −2, the x numerator is −6, and the y numerator is −4.
The displayed solution at two decimal places is x = 3.00, y = 2.00. Checking gives 3 + 2 = 5 and 3 − 2 = 1.
The answer is an ordered pair, so the positions of x and y matter. Geometrically it is the intersection of two lines. Algebraically it is the pair that makes both equalities true at once.
If D is zero, the coefficient rows are proportional. The lines are either distinct and parallel, giving no solution, or they describe the same line, giving infinitely many solutions. The constants decide which case applies, but neither case has one pair for this form to report.
Display precision changes rounding only. When coefficients are measured values or close to proportional, the solution can be sensitive to small input changes. Keep full source precision, then round the final pair appropriately.
Fractions must be entered as decimal values because the fields accept numbers. A repeating decimal is an approximation, so a hand solution with exact fractions may be preferable when exact form matters.
Standardizing the equations is often the hardest part. Moving a term across the equals sign changes its sign. For 2x = 7 − 3y, the standard form is 2x + 3y = 7.
A successful check uses the original equations, not rearranged copies that may contain the same transcription error. Evaluate each left side with the unrounded result where possible and compare it with its constant.
Scaling an entire equation by any nonzero number does not change its line or its solution set. This is a useful check: doubling all three values in one row should leave the final x and y unchanged. Scaling only one coefficient changes the equation and generally changes the answer.
Before calculating, read each row back as a complete equation. This quick check catches misplaced constants, reversed signs and values typed into the wrong coefficient box. If the check fails after solving, return to the original equations rather than adjusting the reported pair.
Each of the two equations needs coefficients for x and y plus a constant. Enter zero for a genuinely missing x or y term.
The lines do not meet once. They may be parallel distinct lines with no common point or the same line with infinitely many common points.
Yes. Preserve each sign and enough decimal precision. Coefficients obtained from measurements may make the final pair approximate.
Use a = 1, b = 1, and c = 5 for that equation. An unwritten coefficient of x or y is one.
No. It only controls displayed decimal places. The calculation uses the submitted coefficient values before output rounding.
Insert the unrounded x and y into both original left sides. Each should equal its own right-side constant within reasonable rounding tolerance.
Substitution and determinant formulas solve the same nonsingular two-equation system. The title describes the learning task; the numerical shortcut provides the same checked pair.