Solve Equations Instantly

Equation Solver helps you solve three types of equations with detailed step-by-step solutions:

Linear Equations

Solve equations in the form ax + b = cx + d, finding the value of x that satisfies the equation.

Quadratic Equations

Solve ax² + bx + c = 0 using the quadratic formula, with discriminant analysis, vertex, and axis of symmetry.

Systems of Equations

Solve systems of 2 or 3 linear equations simultaneously using Cramer's Rule with determinant calculations.

Simply enter your coefficients and get instant results with a complete breakdown of every step in the solving process.

How to Use

Linear Equation Solver

Select Linear Tab

Click on the Linear tab to access the linear equation solver interface.

Enter Coefficients

Input coefficients a, b, c, d for the equation ax + b = cx + d. Leave any field empty to treat it as 0.

View Real-time Preview

The equation preview updates automatically as you type, showing exactly what's being solved.

Get Instant Solution

The solution and step-by-step breakdown appear automatically without pressing any button.

Quadratic Equation Solver

Select Quadratic Tab

Navigate to the Quadratic tab to solve second-degree equations.

Input Coefficients

Enter coefficients a, b, c for the standard form ax² + bx + c = 0.

View Complete Analysis

Instantly see the roots, discriminant (Δ), vertex coordinates, and axis of symmetry.

Follow Step-by-step Solution

Review the detailed solution process using the quadratic formula x = (−b ± √Δ) / 2a.

System of Equations Solver

Select System Tab

Click on the System tab to access the system solver.

Choose Number of Variables

Select 2 vars for two unknowns (x, y) or 3 vars for three unknowns (x, y, z).

Enter Equation Coefficients

Fill in the coefficients for each equation row in the system.

View Cramer's Rule Solution

The solution is calculated automatically using Cramer's Rule with complete determinant calculations.

Features

Three Equation Types

Comprehensive support for multiple equation formats:

Linear equations — Form ax + b = cx + d with one unknown
Quadratic equations — Second-degree using formula x = (−b ± √Δ) / 2a
Systems of equations — 2×2 and 3×3 systems via Cramer's Rule

Step-by-step Solutions

Detailed breakdown of the solving process for every equation type:

Clear identification of coefficients
Each mathematical operation explained
Complete path from input to final answer

Instant Results

Real-time calculation as you type:

No button pressing required
Live equation preview updates
Immediate solution display

Fraction Display

Clean mathematical representation:

Results shown as fractions (e.g. 4/3)
Decimal approximation included
Automatic simplification when possible

Special Case Detection

Automatic identification and explanation of edge cases:

No solution — Contradictory equations
Infinite solutions — Dependent equations
Complex roots — Negative discriminant

Quadratic Properties

Additional analysis for second-degree equations:

Discriminant with color coding (Δ>0, Δ=0, Δ<0)
Vertex coordinates calculation
Axis of symmetry identification

Pro Tip: All coefficient inputs accept decimal numbers, and results will still display as clean fractions when a fractional representation exists.

Frequently Asked Questions

What types of equations can this solver handle?

This tool solves three types:

Linear equations — First degree, one variable (ax + b = cx + d)
Quadratic equations — Second degree, one variable (ax² + bx + c = 0)
Systems of linear equations — 2 or 3 variables solved simultaneously using Cramer's Rule

What is the discriminant?

The discriminant Δ = b² − 4ac determines the nature of a quadratic equation's roots:

Discriminant Value	Root Type	Description
Δ > 0	Two distinct real roots	Equation crosses x-axis twice
Δ = 0	One repeated root	Equation touches x-axis once (vertex on axis)
Δ < 0	Complex roots	No real solutions (parabola doesn't cross x-axis)

What is Cramer's Rule?

Cramer's Rule is a method for solving systems of linear equations using determinants. The process works as follows:

Calculate the main determinant D from the coefficient matrix
For each variable, replace its column with the constants column
Calculate the determinant for each modified matrix (Dx, Dy, Dz)
Divide each variable determinant by the main determinant to get the solution

Formula: x = Dx/D, y = Dy/D, z = Dz/D

When does a system have no solution?

A system has no solution when the equations are inconsistent — they contradict each other.

Mathematical condition: The main determinant D equals 0, but at least one of the variable determinants (Dx, Dy, Dz) is non-zero. This indicates parallel lines (2D) or parallel planes (3D) that never intersect.

What happens when I enter a = 0 in the quadratic solver?

If a = 0, the equation is no longer quadratic — it becomes a linear equation in the form bx + c = 0.

The solver automatically detects this condition and switches to solving it as a linear equation instead, providing the appropriate solution method.

Can I enter decimal numbers?

Yes! All coefficient inputs accept decimal numbers.

Results will still be shown as fractions when a clean fractional representation exists, alongside the decimal approximation for convenience.

Example: Entering 0.5 will display results as 1/2 (0.5) when applicable.