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Elimination Method Calculator

SOLUTION
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Fill in both equations, then press =.
The equations added up, line by line
Type both equations as they are printed — the matching up and the multiplying are done for you.
Left side in the first box, right side in the second. Nothing has to line up first — 3x − 5 = 2y + 1 goes in exactly like that.
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This equation is the one added to or subtracted from the first. Use the same letters in both.
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With three unknowns the elimination runs twice: three equations down to two, then to one.
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Tap a box, then use the keys
The calculator knocks out whichever letter costs the least — a pair that already matches needs no multiplying at all — and says whether to add or subtract, and why. Fractions are cleared first, and the letters need not be x and y.
The elimination, step by step
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This elimination method calculator solves a system of linear equations by cancelling one letter — and writes the sum out the way you would write it on paper.

Take 2x + 3y = 7 with 4x − 3y = 5. The y terms are already a matched pair, one positive and one negative, so nothing needs multiplying: add the two equations straight down the page and the y terms destroy each other, leaving 6x = 12. So x = 2, and putting that back into the first equation gives y = 1.

Elimination Method Calculator — a free tool from Monkza

How the Elimination Method Works

Three moves. Get one letter's coefficients to match in size. Add or subtract the equations so that letter vanishes. Solve what is left, then put the value back into an equation you have not touched.

It is allowed because adding two true equations gives another true equation. Whatever pair of values satisfies both originals must also satisfy their sum, so nothing is lost on the way and nothing is gained. That is also why the method is safe to repeat: every new line you write is still true of the same answer.

Occasionally every letter vanishes at once and only numbers are left. That line is not a mistake — it is the answer. A false line such as 0 = 3 means the equations contradict each other and there is no solution; a true line such as 0 = 0 means one equation simply repeated the other, and infinitely many pairs work.

When to Add and When to Subtract

This one decision is where most marks are lost, and it is settled by the signs alone. Once the two coefficients match in size: opposite signs, add; the same sign, subtract.

There is a way to never have to think about it. If you scale one equation by a negative number instead of a positive one, the coefficients come out opposite rather than equal, and then you always add. Adding is harder to get wrong than subtracting, because subtracting means changing the sign of every term on the second line, including the number on the right — and that forgotten right-hand side is the single commonest slip in the whole method.

In the example above, +3y and −3y are opposite, so adding cancels them. In 2x + 3y = 12 with 3x + 2y = 13, the x terms are +2x and +3x — the same sign — so once they are matched you subtract. The calculator names the decision every time rather than leaving you to spot it.

Matching Coefficients With Multipliers

When no letter already matches, you make it match. Take the lowest common multiple of the two coefficients and scale each equation up to it. For 2 and 3 that is 6, so the first equation is multiplied by 3 and the second by 2:

3 × (2x + 3y = 12) gives 6x + 9y = 36, and 2 × (3x + 2y = 13) gives 6x + 4y = 26. Subtracting leaves 5y = 10, so y = 2, and back in the first equation x = 3.

Those multipliers are the smallest whole numbers that do the job, which keeps the arithmetic small. Fractions are dealt with before any of this: x/2 + y/3 = 4 is multiplied through by 6 to become 3x + 2y = 24, which says exactly the same thing in whole numbers. The answer stays exact either way — that system gives x = y = 24/5.

Elimination With Three Variables

Which pair to combine first is worth a thought. Look for the letter that is already missing from one of the equations, because that equation is half-eliminated before you start — in the system below, the second equation has no x in it at all, so x is the cheapest letter to clear from the other two.

The method does not change, it runs twice. Cancel one letter between two pairs of equations and three unknowns become two; cancel again and you are down to one. Given x + y + z = 6, 2y + 5z = −4 and 2x + 5y − z = 27, the calculator works down to z = −2, then walks back to y = 3 and x = 5, showing each round as its own sum.

How to Use This Elimination Method Calculator

Each equation has two boxes with the equals sign printed between them. Nothing has to be lined up or rearranged first — 3x − 5 = 2y + 1 goes in exactly as printed, and fractions, decimals and brackets are all accepted. The keypad carries a, b and c as well as x, y and z.

Press = and open Show the working. You get the sum laid out: the two equations stacked with their multipliers, a rule underneath, and the line that comes out of adding or subtracting them, with a sentence saying which letter is going and why. Before any answer appears, the values are put back into the equations you typed and both sides are compared exactly — a sign slipped while multiplying is the commonest way this method goes wrong, so it is checked rather than assumed. If you only want the answer and a picture of where the lines meet, the system of equations calculator does that instead.

Elimination Method FAQ

What is the elimination method?
It solves a system by combining the two equations so that one letter cancels. The equations are scaled until that letter's coefficients match, then added or subtracted, which leaves a single unknown to solve directly.
Should I add the equations or subtract them?
Look at the signs of the matched coefficients. If they are opposite, such as +3y and -3y, add. If they agree, such as +3y and +3y, subtract. This calculator states which it is doing and why, because this is where most marks are lost.
When do I have to multiply an equation first?
Only when no letter already has matching coefficients. Take the LCM of the two coefficients and scale each equation up to it. For 2 and 3 the LCM is 6, so one equation is tripled and the other doubled.
Can the elimination method be used with three variables?
Yes, in two rounds. Cancel one letter between two pairs of equations, which leaves two equations in two unknowns, then cancel again. This calculator shows both rounds as separate sums.
What does it mean if every letter disappears when I add?
You are left with numbers only, and that line decides the system. A false line such as 0 = 3 means no solution. A true line such as 0 = 0 means the second equation repeated the first, so infinitely many pairs work.
Why does the calculator clear fractions before it starts?
Because matching coefficients is far easier in whole numbers. Multiplying an equation through by the lowest common denominator changes nothing about what it says, and it keeps every later line tidy.

Match a pair of coefficients, add or subtract, solve what is left, put it back. The whole method is in those four moves — and the signs are the only place it asks you to be careful.