Equation Solver
Enter a single-variable linear equation using standard notation
Example Problems
Enter an Equation to Solve
Select a mode, enter your equation or coefficients, and click Solve to see a step-by-step solution with full working.
How to Use the Equation Solver
Select the Equation Type
Choose from four modes at the top: Linear (one variable), Quadratic (ax² + bx + c = 0), System 2×2 (two equations, two unknowns), or System 3×3 (three equations, three unknowns). Each mode shows tailored input fields for that equation type.
Enter Your Equation or Coefficients
For Linear and System modes, type your equations in standard notation (e.g. 2x + 3y = 7). For Quadratic mode, enter the three coefficients a, b, and c directly in the dedicated fields — no need to type formatted math. Use the example chips to pre-fill a sample problem instantly.
Click Solve and Review Steps
Press the Solve button (or it auto-calculates as you type). The results panel shows the final answer prominently, followed by a numbered step-by-step solution with each algebraic operation labelled. The formula reference card reminds you which formula applies.
Verify and Export
Check the verification row to confirm the solution is correct — it substitutes your answer back into the original equation. Copy the answer to the clipboard with one click, or export all steps to CSV for study notes or further analysis.
Frequently Asked Questions
What types of equations can this solver handle?
This solver handles four categories of equations: single-variable linear equations (e.g. 3x − 2 = 7), quadratic equations in standard form ax² + bx + c = 0 (including those with complex roots), systems of two linear equations in two unknowns (2×2 systems), and systems of three linear equations in three unknowns (3×3 systems). It uses the quadratic formula for quadratics and Gaussian elimination for 3×3 systems. It does not currently support higher-degree polynomials, trigonometric equations, exponential equations, or non-linear systems beyond degree 2.
What is the discriminant and why does it matter?
The discriminant is the expression D = b² − 4ac inside the square root of the quadratic formula. Its value tells you how many real solutions the equation has before you even compute the roots. If D is greater than zero, the equation has two distinct real roots. If D equals zero, there is exactly one real root (a repeated or double root). If D is less than zero, the square root involves the square root of a negative number, which produces two complex conjugate roots of the form a ± bi. Knowing the discriminant first lets you classify the equation type immediately without needing to finish the calculation.
How does the system of equations solver work?
For 2×2 systems, the solver uses the elimination method: it multiplies each equation by the appropriate coefficient to create matching terms for one variable, then subtracts one equation from the other to eliminate that variable and solve for the remaining one. It then back-substitutes to find the second variable. For 3×3 systems, it uses Gaussian elimination on the augmented matrix, applying row operations (swapping rows, scaling, adding multiples) to reach row-echelon form, then back-substitutes from the bottom equation upward. Both methods also detect when the system has no solution (parallel or contradictory equations) or infinitely many solutions (identical equations).
What does it mean when a system has 'no solution' or 'infinitely many solutions'?
A system of two linear equations represents two lines in the plane. If the lines are parallel — same slope but different intercepts — they never intersect, giving no solution (the system is inconsistent). If the lines are identical — one equation is simply a multiple of the other — every point on the line is a solution, giving infinitely many solutions (the system is dependent). When the determinant of the coefficient matrix equals zero, the solver checks both cases and reports the correct classification. A unique solution occurs only when the two lines intersect at exactly one point (the coefficient matrix has a non-zero determinant).
How does the solver show complex roots for quadratics?
When the discriminant D = b² − 4ac is negative, the quadratic equation has no real-number solutions. Instead, the solutions are complex numbers involving the imaginary unit i (where i² = −1). The solver computes the real part −b/(2a) and the imaginary part √|D|/(2a), then displays the two complex conjugate roots in standard a ± bi notation. For example, if a = 1, b = 2, c = 5, the discriminant is 4 − 20 = −16, and the roots are −1 ± 2i. Complex roots always come in conjugate pairs and confirm the equation has no real x-intercepts when graphed as a parabola.
Can I use decimal or negative coefficients for the quadratic mode?
Yes. All three coefficient fields (a, b, c) in quadratic mode accept any real number, including negative values (e.g. a = −2), decimal values (e.g. b = 1.5), and zero for b or c (though a cannot be zero, as that would reduce the equation to linear). For negative coefficients, simply type the minus sign before the number. The quadratic formula works identically regardless of the sign or magnitude of the coefficients. Note that if a is zero, the equation is linear — use Linear mode instead. The decimal-precision selector controls how many digits are shown in the numerical approximation of the roots.