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Equation Solver With StepsOnline Calculator

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Algebra — Standard Rules and Formulas

Quadratic

x=−b±Δ2a,Δ=b2−4acx = \frac{-b \pm \sqrt{\Delta}}{2a}, \quad \Delta = b^2 - 4ac

Δ > 0: 2 real roots | Δ = 0: 1 double root | Δ < 0: 2 complex roots

Cramer's Rule

x=DxD,y=DyD,z=DzDx = \frac{D_x}{D}, \quad y = \frac{D_y}{D}, \quad z = \frac{D_z}{D}

Solves simultaneous linear systems using matrix determinants.

x = ?

Enter expression and press Calculate

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Solving Equations Step by Step

Algebraic equations express equality between two mathematical expressions. By applying balanced operations, we isolate variables and determine exact solutions.

Linear

ax+b=0  ⟹  x=−baax + b = 0 \implies x = -\frac{b}{a}

First-degree equations with a unique real root, infinite solutions, or no solution.

Quadratic

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Second-degree equations analyzed via discriminant Δ for real or complex roots.

Cramer's Rule

xi=det⁡(Ai)det⁡(A)x_i = \frac{\det(A_i)}{\det(A)}

Solves simultaneous linear systems of 2 and 3 variables using matrix determinants.

Discriminant Formula & Nature of Roots

Δ > 0 (Two Real Roots)

The parabola crosses the x-axis at two distinct real points.

x₁, x₂ ∈ ℝ, x₁ ≠ x₂

Δ = 0 (One Double Root)

The vertex of the parabola touches the x-axis at exactly one point.

x₀ = -b / (2a)

Δ < 0 (Complex Conjugates)

The curve does not cross the x-axis; roots are imaginary numbers.

x = u ± vi, i² = -1

Systems of Equations

A system of linear equations is a collection of two or more equations sharing the same set of variables. When solving a 2x2 system, each equation represents a straight line in the Cartesian plane. The solution corresponds to the geometric intersection point of the two lines:

Substitution & Elimination

{a1x+b1y=c1a2x+b2y=c2\begin{cases} a_1 x + b_1 y = c_1 \\ a_2 x + b_2 y = c_2 \end{cases}

Isolating one variable and substituting it into the remaining equations to reduce the dimensions systematically.

Determinants & Matrix Inverse

A⋅x=b  ⟹  x=A−1bA \cdot \mathbf{x} = \mathbf{b} \implies \mathbf{x} = A^{-1} \mathbf{b}

Expressing the linear relations compactly as a matrix equation and inverting the matrix when det(A) ≠ 0.

Frequently Asked Questions

How do you solve an equation with two variables?

An equation with two variables, such as 2x + 3y = 12, has infinitely many solutions, so it does not have a single numerical answer. Solve for one variable in terms of the other: subtract 2x from both sides, then divide by 3 to get y = (12 − 2x) / 3. To find one unique solution, you need a second independent equation, which gives a system of two equations with two variables. Such a system can be solved by substitution, elimination or Cramer's Rule.

What does it mean to solve an equation?

Solving an equation means finding every value of the unknown variable that makes the equation true. For example, x = 3 solves x + 2 = 5 because 3 + 2 = 5. To do this, isolate the variable on one side by applying the same operation to both sides, so the equality stays balanced.

How to solve a multiple variable equation?

Isolate the variable you need and treat the other variables as constants. For example, from V = lwh, you get l = V / (wh). If you need numerical values for several unknowns, you need as many independent equations as unknowns. For three unknowns, use a system of three linear equations, solved with Cramer's Rule, Gaussian elimination or substitution.

How do you solve linear equations step by step?

To solve a linear equation ax + b = c, expand both sides, group all terms containing the variable on the left side, move constant terms to the right side, combine like terms, and divide both sides by the coefficient of x.

How does the discriminant formula work for quadratic equations?

For a quadratic equation ax² + bx + c = 0, the discriminant is Δ = b² - 4ac. If Δ > 0, there are two distinct real solutions; if Δ = 0, there is one double real root; and if Δ < 0, there are two complex conjugate roots involving the imaginary unit i.

What methods are used to solve systems of 2 and 3 equations?

Mathsway uses Cramer's Rule with matrix determinants (D, Dx, Dy, Dz) as well as Gaussian elimination and substitution, providing exact fractional values and step-by-step verification.

What happens when an equation has no solution or infinitely many solutions?

If simplifying an equation leads to a false statement (e.g. 0 = 5), the equation is inconsistent and has no solution (empty set ∅). If simplifying yields an identity (0 = 0), the equation is dependent and has infinitely many solutions (all real numbers ℝ).

Can this equation solver calculate complex or imaginary numbers?

Yes. When a quadratic equation has a negative discriminant (Δ < 0), the calculator displays the exact complex conjugate solutions in the form u ± vi.

Is this equation solver free and private?

Yes, it is 100% free with no signup or login required. All calculations run client-side in your browser using Web Workers, ensuring complete privacy.