Simultaneous Equations Solver

Entering the equations

Every equation goes in as coefficients rather than as typed algebra, so there is nothing to mistype and nothing for the tool to misread. Choose the mode first — two linear equations, three linear equations, or one linear and one quadratic — and the boxes change to match the standard form printed above them.

Whole numbers, decimals and fractions are all accepted, so 3, 2.5 and 3/4 are equally valid, and negatives work anywhere. Arithmetic runs on exact fractions from start to finish, which is why an answer of one third is shown as 1/3 and not as 0.333.

The panel under the inputs restates what you entered as ordinary algebra. Check that line first: if it does not read like the question on the page in front of you, the coefficients went in wrong, and no amount of correct working after that will help.

• Two linear equations — ax + by = c, twice
• Three linear equations — ax + by + cz = d, three times
• A line y = mx + c against a curve y = px² + qx + r
• A line y = mx + c against a circle x² + y² = r²

How the elimination works

To remove an unknown, its coefficient has to be the same size in both equations. The solver finds the smallest whole-number multipliers that achieve that — it uses the lowest common multiple rather than multiplying each equation by the other's coefficient, because the numbers stay smaller and the working stays readable.

Then comes the decision that carries the mark: same sign, subtract; opposite signs, add. The tool states which case it is and what the two matched coefficients are, so the rule is attached to the numbers rather than floating free as something to memorise.

Once one unknown falls out, the second is found by division, substituted back into the first equation to recover the first unknown, and both are then put into the equation that was not used in that substitution. That last line is the check, and it is printed whether or not anyone asked for it.

A three-unknown system is the same idea run twice: x is cleared out of the second and third equations using the first, which leaves two equations in y and z, and those go through exactly the routine above before z and y are carried back into the original equation to find x.

What this solver will not do

It will not silently round. Where an answer is a fraction with a recurring decimal, the exact fraction is given first and the decimal is marked with an ellipsis so that nobody copies a truncated figure into a later part of the question and loses accuracy marks for it.

It will not print no solution for two different situations. Parallel lines and a repeated line both make the determinant zero, but one has no answer at all and the other has infinitely many, and the tool names which by comparing the gradients and the intercepts.

It will not guess when the numbers get too large to stay exact. Rational arithmetic loses precision beyond a certain size, and rather than show a plausible wrong answer, the tool says that the coefficients are out of range and stops.

It also will not solve for you in an exam. The point of reading the working is to be able to reproduce it on paper without the tool, which is the only version of the skill that earns marks.

Where marks are lost

The commonest error by a distance is adding when the signs are the same, or subtracting when they differ. The unknown does not cancel, the arithmetic looks fine, and the answer is wrong from that line onwards — which is why the tool spells out the sign of both matched coefficients before it combines them.

The second commonest is subtracting across the equals sign inconsistently: taking the left-hand sides away from each other but adding the right-hand sides. Writing both scaled equations out in full, as the working here does, makes that slip visible.

In the one-linear-one-quadratic case, students often find both values of x and stop, forgetting that each x has a y with it and that the question asks for points. Two roots means two coordinate pairs, and half the marks sit in the second one.