Simultaneous equations solver

Enter two equations and this solves them by elimination, showing every step including which variable it chose to eliminate and why. Also called systems of equations in US courses.

Equation 1 x + y =

Equation 2 x + y =

Both rows are written with a plus sign, so type negative coefficients directly. To enter 4x − y = 5, put 4, then −1, then 5.

What you are actually finding

Each equation describes a straight line. Solving them simultaneously means finding the point where those two lines cross, which is the one pair of x and y values that satisfies both equations at the same time.

That picture explains the odd cases too. Two parallel lines never cross, so there is no solution. Two equations describing the same line cross everywhere, so there are infinitely many solutions. Both are covered below.

Which method should you use? Elimination is usually faster when the coefficients are whole numbers, which is most school questions. Substitution is easier when one equation already has a variable on its own, like y = 3x + 1. Both give the same answer, so use whichever the question sets up more neatly.

How elimination works

The idea is to make one variable disappear. If both equations have the same number of y terms, subtracting one equation from the other removes y entirely and leaves a simple equation in x alone.

Usually the coefficients do not match to begin with, so you multiply one or both equations by whatever makes them match. Multiplying an entire equation by a number does not change what it means, since you are doing the same thing to both sides.

Once you have one variable, substitute it back into either original equation to get the other. Then check your answer in the equation you did not use, which catches arithmetic slips.

Common questions

Are simultaneous equations the same as systems of equations?: Yes. UK, Australian and Indian courses generally say simultaneous equations; US courses usually say systems of equations. Same topic, same methods.
When do I add rather than subtract?: Look at the signs of the matching terms. If they are the same sign, subtract to cancel them. If they are opposite signs, adding cancels them instead. Getting this backwards is the most common slip in elimination.
What does "no solution" mean in a real question?: That the two conditions genuinely contradict each other. In a word problem it usually means the situation as described is impossible, so it is worth rechecking how you translated the words into equations.
Can I check my answer?: Yes, and you should. Put both values back into the equation you did not use for the substitution step. If it balances, your answer is right. This tool does that check automatically and shows it.