Elimination Calculator - Solve Systems by Elimination, Step by Step

Put both equations in standard form first: $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$. Then fill the six boxes of the elimination calculator and click Calculate.

• $a_1$, $b_1$, $c_1$ — Equation 1: the x-coefficient, the y-coefficient, and the right side. For 3x − 2y = 7 they are 3, −2, 7 — the elimination calculator's default system.
• $a_2$, $b_2$, $c_2$ — the same three numbers for Equation 2. Keep the sign as it stands: in 8x + 2y = 48, the y-coefficient $b_2$ is 2.
• Check — the elimination calculator's guard row: it tests whether the determinant $a_1b_2 - b_1a_2$ is nonzero. True means exactly one solution.
• x and y — the solution pair, rounded to four decimals. The elimination calculator prints one row per letter.

The elimination calculator starts pre-filled with 3x − 2y = 7 and 8x + 2y = 48; one click solves it: x = 5, y = 4. Swap in your own numbers and the elimination calculator re-solves instantly.

A false Check row means the determinant is 0: no solution or infinitely many. The special-cases table below separates them in seconds, so the elimination calculator never leaves you guessing.

This page is more than a static elimination calculator — it is also an AI tutor. The box under the form takes full questions, photos, and follow-ups:

• Ask in plain words. "My equations are 2x + 3y = 12 and x − y = 1 — solve the system by elimination." The AI solves the system step by step right in the chat.
• Take a photo of your homework. The AI reads the system and solves it step by step.
• Ask about the ideas, not just the answer. "Why do the y terms cancel?" The AI explains step by step; you can check every number against the calculator on this page.
• Work backwards. Give a solution point and ask which coefficients produce it. The AI walks it backwards with you, then the elimination calculator confirms the pair.
• Keep asking follow-ups. Ask "why?" as often as you need; the elimination calculator page keeps teaching.

Elimination means combining two whole equations so one letter vanishes. If the y-coefficients are −2 and 2, adding the equations kills both y terms in a single stroke. One equation, one letter — and solving it is routine. That combination is the one move an elimination calculator makes, over and over. The elimination calculator just never gets tired of it.

A 19th-century school algebra defines the operation: combine two equations in two unknowns and deduce from them a single equation in one unknown. That definition still drives every elimination calculator, this one included. This elimination calculator runs it on whatever pair you type.

Read the picture: the blue line is every pair with 3x − 2y = 7, the orange line every pair with 8x + 2y = 48. The lines cross exactly once, at (5, 4). Every elimination calculator question ends at a crossing like this one. One crossing, one solution — that is the elimination calculator's Check row in graphical form.

Here is the full routine — you can solve the system by elimination by hand in seven short steps:

1. Write both equations in standard form; clear any fractions first.
2. Multiply one or both equations so a pair of coefficients becomes opposites.
3. Add the equations and watch that letter cancel.
4. Solve the one-letter equation that remains.
5. Substitute back into either original equation for the second letter.
6. Write the answer as an ordered pair (x, y).
7. Check the pair in both original equations.

Every example below runs these steps; the elimination calculator collapses them into one click.

Hand elimination always ends at the same two fractions, so the elimination calculator can evaluate them directly:

$x = \dfrac{c_1b_2 - b_1c_2}{a_1b_2 - b_1a_2} \qquad y = \dfrac{a_1c_2 - c_1a_2}{a_1b_2 - b_1a_2}$

Both share one denominator — the determinant:

$D = a_1b_2 - b_1a_2$

The Check row of the elimination calculator computes exactly this number. A zero there is the elimination calculator's only stop sign.

Special cases — how the elimination calculator reports a false Check row:

Try 4x + 6y = 10 with 2x + 3y = 5: every ratio matches, one line drawn twice, infinitely many solutions. Change $c_1$ to 12 and no pair works: no solution. The elimination calculator flags both with a false Check row; the ratios tell you which.

Solve the anchor system by elimination, one line per step — the elimination calculator's own default:

1. Look at the y-coefficients: −2 and 2. They are opposites already, so no multiplying.
2. Add the two equations member to member: $11x = 55$.
3. Divide: $x = 5$.
4. Substitute back into Equation 1: $15 - 2y = 7$, so $y = 4$.
5. Check both equations: $3 \cdot 5 - 2 \cdot 4 = 7$ and $8 \cdot 5 + 2 \cdot 4 = 48$. The solution is $(5, 4)$.

The form above holds exactly this system: $a_1 = 3$, $b_1 = −2$, $c_1 = 7$, $a_2 = 8$, $b_2 = 2$, $c_2 = 48$. Click Calculate and the elimination calculator confirms x = 5, y = 4 in one run. That is the elimination calculator's whole promise: six coefficients in, two values out.

When two matching coefficients carry the same sign, subtract instead of add:

1. Line up 8x + 2y = 30 and 3x + 2y = 15: the y-coefficients are 2 and 2 — identical, so no multiplying.
2. Subtract the second equation from the first: $8x - 3x = 5x$ and $30 - 15 = 15$, so $5x = 15$.
3. Divide: $x = 3$.
4. Substitute back: $24 + 2y = 30$, so $y = 3$.
5. Check both equations: $24 + 6 = 30$ and $9 + 6 = 15$. The solution is $(3, 3)$.

Type 8, 2, 30 and 3, 2, 15 into the elimination calculator above. The elimination calculator's x and y rows return 3 and 3 without a single fraction. Same sign means subtract; opposite signs mean add — the elimination calculator does the bookkeeping either way.

Sometimes the solution is given and a coefficient is missing — work the elimination calculator's direction in reverse. Suppose x + y = 10 and 3x − by = 35 have the solution (11, −1). Find b.

1. A solution must satisfy both equations, so feed x = 11 and y = −1 to the second one.
2. That gives $33 + b = 35$.
3. Solve: $b = 2$.
4. The full system is x + y = 10 with 3x − 2y = 35, and indeed $11 - 1 = 10$ while $33 + 2 = 35$.

With b = 2 the determinant is $1 \cdot (−2) - 1 \cdot 3 = −5$, not 0. One solution exists, and the elimination calculator reports it as x = 11, y = −1 — run it and see. Change b to 4 and the elimination calculator returns (10.7143, −0.7143) instead: the system itself changed.

Three slips the elimination calculator catches every day:

1. Multiplying part of an equation. Doubling 3x − 2y = 7 must give 6x − 4y = 14, not 6x − 2y = 7. Every term on both sides gets the multiplier; the elimination calculator's x and y rows keep the algebra honest.
2. Adding coefficients that are not opposites. Adding is only a cancel when the signs differ. With 2y and 2y you subtract; with −2y and 2y you add. Decide by sign before you combine, or the elimination calculator will happily solve a system you did not mean to write.
3. Misreading a false Check row. A zero determinant is not an error message. 0 = 0 means one line drawn twice — infinitely many solutions. 0 = 6 means parallel lines — no solution. The ratios table above separates the two; the elimination calculator flags both the same way.

Name the slip, fix it, and the elimination calculator confirms the repair — solve one by hand for practice.