๐งฎ System of Equations Solver
Two equations, two unknowns, every elimination step shown.
Equation 1: aโx + bโy = cโ
Equation 2: aโx + bโy = cโ
Solution (x, y)
How to Use the System Solver
Write each equation in the form ax + by = c and enter the six coefficients. The solver uses elimination, shows every step, and flags parallel or identical lines.
Rearrange each equation to ax + by = c form if needed.
Enter the six coefficients aโ, bโ, cโ, aโ, bโ, cโ.
Read the solution and verify it in both original equations.
Elimination Method
Elimination cancels one variable by adding or subtracting scaled equations. For 3x + 2y = 12 and x โ y = 1: multiply the second by 2 to get 2x โ 2y = 2, add to the first: 5x = 14, so x = 2.8. Substitute back: 2.8 โ y = 1, so y = 1.8. The determinant D = aโbโ โ aโbโ predicts the outcome before you start: D โ 0 means one unique solution; D = 0 means parallel (no solution) or identical (infinite solutions).
Substitution Method
Solve one equation for one variable, then plug into the other. From x โ y = 1, x = y + 1. Substitute into 3x + 2y = 12: 3(y+1) + 2y = 12, so 5y = 9, y = 1.8, x = 2.8. Substitution is cleanest when a coefficient is 1 or โ1. Both methods give the same answer; use whichever keeps the arithmetic simplest.
Three Possible Outcomes
One solution: lines cross at a single point (D โ 0). Most textbook problems. No solution: parallel lines (D = 0 but equations inconsistent, e.g. 0 = 5). Real-world: two pricing plans that never break even. Infinite solutions: same line twice (D = 0 and consistent, e.g. 0 = 0). The equations were redundant. Recognizing which case you are in before grinding arithmetic saves real time on exams.
Frequently Asked Questions
What is a system of equations?
Two or more equations sharing the same unknowns, solved together. The solution is the point (or points) satisfying all equations at once. Graphically, each linear equation is a line, and the solution is where the lines cross.
Substitution or elimination: which is better?
Both always work; pick by convenience. Substitution shines when one variable is already isolated (y = 2x + 1). Elimination shines when coefficients line up for easy cancellation (3x + 2y = 12 and 3x โ 2y = 4: add them). This solver shows the elimination path step by step.
What does "no solution" mean?
The lines are parallel: same slope, different intercept. Algebraically, elimination produces a false statement like 0 = 5. Example: x + y = 3 and x + y = 7 can never both be true. The solver detects this automatically.
What does "infinitely many solutions" mean?
The equations describe the same line: one is a multiple of the other. Elimination produces 0 = 0, true for every point on the line. Example: 2x + 2y = 6 and x + y = 3. Every point on the line y = 3 โ x is a solution.
How do I check my answer?
Plug the (x, y) values back into BOTH original equations. Each must balance. For x = 2, y = 1 in 3x + 2y = 8: 6 + 2 = 8 โ. A solution that fails either equation is wrong, usually from an arithmetic slip in the elimination step.