Equation Solver Step by Step — Linear, Quadratic, Systems Guide

Equation Types Reference

How to Use the Equation Solver

1. Open the equation solver.
2. Select the equation type: Linear, Quadratic, or System (2 equations).
3. Enter the coefficients in the input fields (negative values accepted).
4. Click Solve Step by Step.
5. Follow each step from the starting equation to the final answer.

Solving Linear Equations

A linear equation has the form ax + b = c. The goal is to isolate x. The two steps are: (1) subtract b from both sides to get ax = c − b, and (2) divide both sides by a to get x = (c − b) / a. Special cases: if a = 0 and b = c, the equation is an identity (true for all x); if a = 0 and b ≠ c, there is no solution.

Solving Quadratic Equations

The quadratic formula

For ax² + bx + c = 0, the quadratic formula is x = (−b ± √(b² − 4ac)) / 2a. The ± gives two solutions (if the discriminant is positive), one solution (if zero), or complex solutions (if negative).

The discriminant

The discriminant Δ = b² − 4ac determines the nature of the roots before solving:

• Δ > 0 — two distinct real roots
• Δ = 0 — one repeated root (the parabola touches but does not cross the x-axis)
• Δ < 0 — no real roots (the parabola is entirely above or below the x-axis)

Factoring versus the formula

When Δ is a perfect square, the quadratic factors neatly (e.g. x² − 5x + 6 = (x − 2)(x − 3)). When Δ is not a perfect square, factoring is impractical and the formula gives exact irrational roots. The solver always uses the formula for generality and shows the discriminant calculation first.

Solving Simultaneous Equations

Cramer's rule

For the system a₁x + b₁y = c₁ and a₂x + b₂y = c₂, Cramer's rule uses determinants: x = (c₁b₂ − c₂b₁) / (a₁b₂ − a₂b₁) and y = (a₁c₂ − a₂c₁) / (a₁b₂ − a₂b₁). If the denominator equals zero, the system has either no solution (inconsistent) or infinitely many solutions (dependent).

Substitution method

An alternative to Cramer's rule is substitution: rearrange one equation for one variable, substitute into the other, then back-substitute to find the first. Cramer's rule is more systematic for 2×2 systems; substitution is more transparent for simpler systems.

Checking Solutions

Always verify answers by substituting back into the original equation. For x = 4 in 2x + 3 = 11: 2(4) + 3 = 11. For x = 2, y = 1 in 2x + y = 5: 2(2) + 1 = 5. If both sides match, the solution is correct.