Systems of Equations Calculator

One row per equation, one click. This systems of equations calculator turns each equation into one row of a grid, so you never touch matrix notation yourself - the page keeps the bookkeeping. Pick the size first - the grid starts as a 2x2 system and stretches to 3x3, so one grid covers both homework sizes.

• Each row holds one equation written as $ax + by = c$: the coefficients of x, then y, then z for a 3x3. The right side goes in the last column - the shape every systems of equations calculator expects.
• The systems of equations calculator preloads the system x + y = 10 and 2x - y = 5. Click Calculate as is, or type your own six numbers into the grid and watch it update in one click.

Two results come back from the systems of equations calculator, and both matter:

• Determinant of the coefficient matrix - one number, and it must not be 0. For the defaults the systems of equations calculator reads -3, which promises exactly one solution.
• Solution vector - printed like [[5],[5]]. Read it top to bottom: x first, then y (then z for a 3x3). Here the systems of equations calculator reports x = 5 and y = 5.

Under the hood, this systems of equations calculator eliminates variables with an LU factorization and rounds to 10 digits. That is why it stays exact where hand arithmetic drifts. The defaults 1, 1, 10, 2, -1, 5 come back clean.

This page is more than a static systems of equations calculator - it is also an AI tutor you can ask questions:

• Ask in plain words. "One taco and one juice cost 10 dollars; two tacos and minus one juice cost 5 - solve it." The AI solves it step by step right in the chat - the calculator does every bit of the arithmetic.
• Take a photo of your homework. The AI reads the equations and hands them to the systems of equations calculator form, no retyping needed.
• Ask about the ideas, not just the number. "Why does adding the equations kill y?" The AI explains, and you can check every number against the systems of equations calculator on this page.
• Go bigger. Switch the row count to 3 and the same form becomes a 3x3 system solver - Example 2 walks one fully; the systems of equations calculator checks it in one click.
• Keep asking follow-ups. Ask "why?" as often as you need - this systems of equations calculator page keeps teaching long after the boxes are empty.

Take the default system the systems of equations calculator preloads: x + y = 10 and 2x - y = 5. Each line is one condition, and a solution must satisfy both at the same time. That is what a simultaneous equation solver is for, and what a systems of equations calculator computes. The picture shows the two conditions as two lines, and the solution is the one point where both hold: their crossing (5, 5). That point is what a systems of equations calculator is built to find.

$\begin{cases} x + y = 10 \\ 2x - y = 5 \end{cases}$

That crossing point is all a systems of equations calculator looks for - it is the one answer a systems of equations calculator exists to give. Check it by hand: 5 + 5 = 10 and 2(5) - 5 = 5 - the same two-equation check the systems of equations calculator automates. Both equations agree, so (x, y) = (5, 5) is the solution, and it is the only one - the very claim the systems of equations calculator will certify. The systems of equations calculator finds the same pair with algebra alone - the graph is just the receipt, and a systems of equations calculator trades the picture for exact numbers.

When does a system have exactly one solution like this? Exactly when the two lines are not parallel, and the determinant tests that before the systems of equations calculator solves anything. For a 2x2 system $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$, the determinant is

$D = a_1b_2 - a_2b_1.$

For the defaults, D = (1)(-1) - (2)(1) = -3, nonzero - so one crossing point exists, and the systems of equations calculator confirms it in its first output row. The variables match the grid columns one to one: x in column 1, y in column 2, the right side c in the last column. The systems of equations calculator prints the same layout. Example 3 shows what happens when D = 0 - the one case where no systems of equations calculator can return a single pair.

Classrooms teach three routes to the same crossing point. Each one is a solving method the systems of equations calculator can retrace for you, so pick one and let the systems of equations calculator check the landing:

Before you solve the system of equations by elimination, line both equations up in the same order - x, then y, then the constant. Line them up wrong and the columns stop matching, so the systems of equations calculator would solve a different system than yours. To solve a system of equations by substitution instead, isolate the easiest variable, then substitute its expression into the other equation. The systems of equations calculator accepts the result either way. Pick whichever feels natural - it lands on the same pair regardless. A systems of equations calculator does not mind which route you drove; it checks the destination.

Whichever route you pick, the systems of equations calculator runs the elimination bookkeeping and hands back the exact pair. That is the promise of a systems of equations calculator: exact where graphing only estimates. It is a fast system solver for checking homework. It is a systems of equations solver for the messy fractional cases where graphing fails. And it stays a patient tutor when you want the why behind each move of a systems of equations calculator. For the full row-operations method, see the Gaussian elimination calculator - it is the same augmented-matrix work, one column at a time.

Solve the system 2x + y = 7 and x - 2y = 6. It is a home game for the systems of equations calculator - type it in and the systems of equations calculator checks every step below in one click.

1. Line up the equations. Column x holds 2 and 1; column y holds 1 and -2 - the same two columns the systems of equations calculator shows.
2. The y-coefficients are 1 and -2. Multiply the first equation by 2: 4x + 2y = 14.
3. Add it to the second equation: 5x = 20, so x = 4 (the systems of equations calculator does this addition entry by entry).
4. Substitute back: 4 - 2y = 6, so -2y = 2 and y = -1.

The solution is (x, y) = (4, -1). Check the untouched originals: 2(4) + (-1) = 7 and 4 - 2(-1) = 6 - the two checks the systems of equations calculator stands behind. Both hold - and the systems of equations calculator agrees: determinant D = (2)(-2) - (1)(1) = -5, solution [[4],[-1]].

Notice the determinant worked before any arithmetic: D = -5 ≠ 0 promised a unique answer. Type 2, 1, 7, 1, -2, 6 into the grid and the systems of equations calculator returns exactly this pair. That is the systems of equations calculator keeping its side of the deal - two outputs, one story, no arithmetic for you to trust on faith. For a model of how a systems of equations calculator thinks, this pair is it: eliminate, back-substitute, verify. A system of equation solver is most trustworthy when its two outputs tell the same story - here the nonzero determinant and the clean pair do. This systems of equations calculator shows both.

Solve this system of equations in three unknowns - the size where a systems of equations calculator really earns its keep:

$\begin{cases} x + y + z = 6 \\ 2x - y + z = 3 \\ x + 2y - z = 2 \end{cases}$

1. Eliminate z from equations 1 and 3: add them, 2x + 3y = 8 - the same first move the systems of equations calculator makes internally.
2. Eliminate z from equations 1 and 2: subtract, x - 2y = -3.
3. Now eliminate x: multiply x - 2y = -3 by 2 and add, giving -y = -2, so y = 2.
4. Back-substitute: x = -3 + 2(2) = 1, then z = 6 - 1 - 2 = 3 - the order in which the systems of equations calculator reads its own vector.

The solution is (x, y, z) = (1, 2, 3) - the same triple the systems of equations calculator prints as [[1],[2],[3]]. Verify in all three originals the way the systems of equations calculator does: 1 + 2 + 3 = 6, 2(1) - 2 + 3 = 3, 1 + 4 - 3 = 2. All three hold - exactly as the systems of equations calculator insists.

The 3x3 determinant is 7, nonzero, so the unique answer was guaranteed before elimination began - a guarantee the systems of equations calculator prints before it solves. Switch the grid to 3x3 and type the twelve entries into the systems of equations calculator on this page. The systems of equations calculator returns det = 7 and the vector [[1],[2],[3]] in one click - the whole point of a systems of equations calculator. Solve systems of equations like this one by clearing one column at a time, and the systems of equations calculator will mirror every move.

Work backwards for once: from a dead determinant to the whole solution set. This is the case every systems of equations calculator must explain clearly.

No solution. Take x + 3y = 4 and -2x - 6y = 3 - the classic case a systems of equations calculator must decline politely. The determinant is D = (1)(-6) - (-2)(3) = 0, the first warning the systems of equations calculator prints in its determinant row. Doubling the first equation gives 2x + 6y = 8; adding the second gives 0 = 11 - a false statement. The two lines are parallel. No crossing point exists, so the systems of equations calculator correctly refuses to print a vector - there is nothing for a systems of equations calculator to find.

Infinitely many. Now change the same system to x + 2y = 4 and 2x + 4y = 8 - the systems of equations calculator returns the same dead determinant for it. Again D = (1)(4) - (2)(2) = 0. But doubling the first equation now gives exactly the second - both equations describe one identical line. Every point on x + 2y = 4 solves the system: (0, 2), (4, 0), (2, 1), and infinitely many more. A solution set that large is exactly why no systems of equations calculator prints it one vector at a time.

Same dead determinant, two different stories - the systems of equations calculator shows det = 0 for both. The tiebreak is the row of zeros after elimination: 0 = 0 means infinitely many solutions, while 0 = nonzero means none. This systems of equations calculator trains you to make that reading. When the systems of equations calculator prints det = 0 it is not failing - det = 0 is the systems of equations calculator telling the truth about your system. It is reporting that no single vector exists for any systems of equations calculator to give - and the two cases stay easy to tell apart by hand.

Four slips this systems of equations calculator catches every day:

1. Columns out of order. x-coefficients go in column 1, y-coefficients in column 2, constants last. Line both equations up as $ax + by = c$ before typing, or the systems of equations calculator solves a different system than the one on your homework.
2. Scaling one side only. Multiplying an equation by 2 must hit both sides, constants included. 2x + y = 7 becomes 4x + 2y = 14 - not 4x + 2y = 7. Half an equation is the classic silent error - no systems of equations calculator can fix a typo it was never given.
3. Dropping minus signs on substitution. From x - 2y = 6, substituting x = 6 + 2y (not 6 - 2y) is where most points are lost. Slow down at every negative.
4. Calling det = 0 a bug. A zero determinant means no unique solution exists - the system has infinitely many solutions or none, and this systems of equations calculator says so instead of guessing. Example 3 gives the two-minute tiebreak.

Type carefully and the systems of equations calculator does the rest - it never tires of the arithmetic you do.