Elimination Method Calculator

What this calculator does

The elimination method is the technique most of us learn first for solving two equations at once: line them up and add or subtract them so that one of the variables disappears, leaving something you can solve straight away. This applies that method to a two-variable system.

Enter the coefficients of both equations and it returns the solution. It runs right here in the browser.

Using the calculator

1. Enter the coefficients and constant for the first equation (a1, b1, c1).
2. Enter them for the second equation (a2, b2, c2).
3. Press Calculate.

It returns the values of x and y. Reset clears everything.

What the elimination method is

The whole trick is to get rid of one variable so you are left with an easy equation in the other. If the two equations have opposite amounts of, say, y (a plus 3y in one and a minus 3y in the other), then simply adding the equations makes the y terms cancel out. If they have the same amount of y, then subtracting one from the other cancels it instead. Either way, y vanishes, and you are left with a single equation in x alone.

Once you have solved that for x, you slot the value back into either original equation to find y. Two unknowns become one, then none.

The steps

1. Match a variable. If neither variable already has matching or opposite coefficients, multiply one whole equation (or both) by a number so that one variable does.
2. Add or subtract the equations to eliminate that variable.
3. Solve the resulting single-variable equation.
4. Substitute back into an original equation to find the other variable.

Step 1 is the one people skip and then get stuck on. When the coefficients do not line up on their own, a quick multiplication to make them match is what makes the cancelling possible.

A worked example

Solve 2x + 3y = 8 and x + y = 3. Neither variable matches yet, so start with step 1.

1. Multiply the second equation by 2 so its x term matches the first: 2x + 2y = 6.
2. Subtract this from the first equation: (2x + 3y) − (2x + 2y) = 8 − 6, which leaves y = 2. The x terms cancelled, as planned.
3. Substitute into x + y = 3: x + 2 = 3, so x = 1.

The solution is x = 1, y = 2. Check both: 2(1) + 3(2) = 8, and 1 + 2 = 3. Both hold.

No solution, or infinitely many

Elimination does something helpful when a system is not straightforward, and this tool reports it. If, when you try to eliminate a variable, both variables vanish and you are left with a true statement like 0 = 0, the two equations are really the same line, so there are infinitely many solutions. If instead you are left with a false statement like 0 = 5, the equations contradict each other, describing parallel lines, so there is no solution at all. The method does not just fail quietly in these cases, it tells you which situation you are in.

Elimination versus the other methods

Elimination is one of a small family of ways to solve a system. Its closest cousin is substitution, where you solve one equation for a variable and plug it into the other, which is often tidier when a coefficient is already 1. For three variables or a general solve, the system of equations calculator uses Gaussian elimination, which is simply this same idea carried out systematically on larger systems. And for other routes entirely there are Cramer's rule with determinants and reduced row echelon form. All arrive at the same answer.

Questions people ask

What is the elimination method?

A way to solve two equations by adding or subtracting them so one variable cancels, leaving a single equation to solve, then substituting back for the other.

When do I need to multiply an equation first?

When neither variable has matching or opposite coefficients. Multiplying an equation scales its coefficients so one variable lines up and can be cancelled.

Do I add or subtract the equations?

Add when the coefficients of the variable are opposite, subtract when they are the same. Either way, that variable disappears.

What if there is no unique solution?

If both variables cancel to leave a true statement, there are infinitely many solutions. If they leave a false one, there is no solution.

How is it different from substitution?

Substitution solves one equation for a variable and inserts it into the other. Elimination combines the equations to cancel a variable. Both work; the tidier choice depends on the numbers.

References

A note on where this comes from. The elimination method solves a linear system by combining equations to cancel variables. Carried out systematically on larger systems it becomes Gaussian elimination, named after Carl Friedrich Gauss, though the technique appears much earlier in the ancient Chinese Nine Chapters on the Mathematical Art. For further reading, see System of linear equations.

1. The elimination method, combining equations by addition or subtraction to remove a variable.
2. Gaussian elimination, the systematic extension of the method to larger systems.