Systems of equations, row by row

You get exact answers where they exist, like (−3 ± √17)/4 or 1 ± 2i, and the working in plain language: the quadratic formula, rational roots, row operations or Newton’s method. Free and private, with step-by-step working.

How to solve an equation

1. Type an equation, or a system with one equation per line, into Equation or system. Press Try one for an example.
2. Under Options, set the range for Search decimals from and to, and choose whether Angles in degrees.
3. Press Solve.
4. Read the answers and the working. Press Copy the working to take it with you.

What you can solve

Linear equations like 3(x − 1) = 2x + 4, quadratics, cubics and polynomials up to degree 24, with real and complex roots. Systems of linear equations with two to six unknowns, written one per line. And single-unknown equations with functions, such as cos(x) = x or x^2 = 2^x, which are solved to about twelve significant figures.

How the working is done

Linear equations are rearranged step by step. Quadratics use the formula, with the discriminant shown and square roots simplified, so you see √8 become 2√2. For cubics and higher, the solver tests every possible rational root p/q, divides out each one it finds, and finishes with the quadratic formula when it can. Systems are written as an augmented matrix and reduced with row operations in exact fractions, then checked by substituting back.

When the answer is a decimal

Some roots have no tidy exact form, like the real root of x³ = 2. Those are found numerically and marked with ≈. Equations with sin, log or roots are scanned for sign changes between the two search limits, then each root is pinned down with Newton’s method, falling back to halving the interval when a step jumps too far. If you expect a root outside the range, widen it. Everything runs in your browser and nothing you type is sent anywhere.

Tips for entering equations

• Use ^ for powers, such as x^2, and brackets to make the order clear.
• Write one equation per line for systems, using the same letters for the unknowns.
• Check the angle setting before solving anything with sin, cos or tan.
• Verify an answer by putting it back into the original equation.

Who uses it

Students checking homework and learning the steps, parents helping with maths, and tradespeople or hobbyists who need a quick answer. To see an equation as a picture, use the graphing calculator. For fractions, try the fraction calculator.

Questions

Why do I get complex answers?

Some equations have no real solution, and the answers involve i, the square root of minus one. Many courses expect this for quadratics with a negative discriminant.

Can it solve every equation?

No. It handles polynomials, linear systems and single-unknown equations with common functions. Equations with several unknowns that are not linear are not supported.

Is it accurate?

Exact answers are shown where they exist. Numerical answers are worked to about twelve significant figures.