Gaussian Elimination Calculator

One row per equation, one click. This gaussian elimination calculator takes each equation as one row of an augmented matrix, so you never touch matrix notation yourself. Pick the size first - the gaussian elimination calculator grid adjusts from 2 equations up to 6.

• Each row holds one equation: the coefficients of x, then y, then z for a 3x3, with the right side behind the bar.
• The gaussian elimination calculator preloads 2x + y = 5 and x - y = 1. Click Calculate, or change the row count for bigger systems.

Both rows together form the augmented matrix that the gaussian elimination calculator reduces.

Two results come back from the gaussian elimination calculator, and both matter:

• Determinant of the coefficient matrix - one number, and it must not be 0. For the defaults the gaussian elimination calculator reads -3, which promises exactly one solution from this gaussian elimination calculator run.
• Solution vector - printed like [[2],[1]]. Read it top to bottom: x first, then y (then z and beyond for larger grids). Here x = 2 and y = 1.

Under the hood, this gaussian elimination calculator runs the elimination as an LU factorization, then rounds to 10 digits. The defaults 2, 1, 5, 1, -1, 1 come back clean - the gaussian elimination calculator rounds the noise away. When the determinant reads 0, the solution row has nothing unique to print. Example 3 shows how the gaussian elimination calculator behaves there.

This page is more than a static gaussian elimination calculator - it is also an AI tutor you can ask questions:

• Ask in plain words. "Two tacos and one juice cost 5 dollars; one taco and minus one juice cost 1 - solve it." The AI fills the six boxes and runs the calculator on this page for you - the gaussian elimination calculator with zero typing.
• Take a photo of your homework. The AI reads the equations and fills the gaussian elimination calculator form.
• Ask about the ideas, not just the number. "Why does adding one row to another eliminate x?" The AI explains each row operation, and every number still comes from the calculator on this page.
• Go bigger. Need a gaussian elimination method solver for three, four, even six equations? Switch the row count and the same form solves them all - Example 2 walks a full 3x3, and the AI retraces any step with you.
• Keep asking follow-ups. Ask "why?" as often as you need - this gaussian elimination calculator page keeps teaching long after the boxes are empty.

Take the default system 2x + y = 5 and x - y = 1. Stack the coefficients in a grid and keep the constants behind a bar. The equal signs turn into one vertical line - the gaussian elimination calculator's starting grid:

That bracketed grid is the augmented matrix - the object this gaussian elimination calculator actually works on. Each row is still an equation, and each column belongs to one variable - all the structure the gaussian elimination calculator needs. A missing term enters as 0.

Gaussian elimination cleans the grid with row operations until the answer reads off the last column - the gaussian elimination calculator's whole job. A 1904 college algebra text puts the goal plainly: elimination derives, from two equations, a new one where an unknown disappears. The gaussian elimination calculator performs that disappearance one whole column at a time.

The picture shows why the arithmetic cannot fail when the determinant is nonzero. The two rows are two lines, and the solution is their crossing point (2, 1). The gaussian elimination calculator finds that same point with algebra alone - for a gaussian elimination calculator, the graph is just the receipt.

How to solve system of equations with row operations

Only three moves are legal, and the gaussian elimination calculator uses no others:

The target shape is row-echelon form: 1s down the main diagonal, 0s everywhere below. Gaussian elimination stops there and back-substitutes. Gauss-Jordan elimination sweeps above the diagonal too, until the last column is the answer itself. That final shape is what the gaussian elimination calculator aims past.

The grid of this gaussian elimination calculator is exactly the augmented matrix: one row per equation, one column per variable, the constants in the last column. For the default 2x2 that is $a_1, b_1 | c_1$ over $a_2, b_2 | c_2$.

The determinant row computes $D = a_1b_2 - b_1a_2$ (the n×n generalization runs the same way), the gaussian elimination calculator's first printed result. Line both equations up as $ax + by = c$ first, so each variable keeps its own column - the gaussian elimination calculator trusts your typing.

How to do elimination in algebra without the brackets

Multiply one equation until an x-coefficient matches the other, then add or subtract the equations whole. An 1859 text lists this as elimination by addition or subtraction, its first method. The rows of the gaussian elimination calculator keep the identical bookkeeping. This gaussian elimination calculator carries the method forward - the brackets just hold your place.

Solve 2x + y = 5 and x - y = 1, the exact system this gaussian elimination calculator preloads as placeholders. It is the gaussian elimination calculator's home game, and all three row operations get used once.

1. Write the augmented matrix: $\left[\begin{array}{cc|c} 2 & 1 & 5 \\ 1 & -1 & 1 \end{array}\right]$
2. Swap for a leading 1, $R_1 \leftrightarrow R_2$: $\left[\begin{array}{cc|c} 1 & -1 & 1 \\ 2 & 1 & 5 \end{array}\right]$
3. Kill the x below it, $R_2 \leftarrow R_2 - 2R_1$: $\left[\begin{array}{cc|c} 1 & -1 & 1 \\ 0 & 3 & 3 \end{array}\right]$
4. Scale the second row, $R_2 \leftarrow \tfrac{1}{3}R_2$: $\left[\begin{array}{cc|c} 1 & -1 & 1 \\ 0 & 1 & 1 \end{array}\right]$
5. Back-substitute: row 2 reads y = 1, then row 1 gives x - 1 = 1, so x = 2.

This is the shape the gaussian elimination calculator reports on.

Check both original equations: 2(2) + 1 = 5 and 2 - 1 = 1. The pair (x, y) = (2, 1) works everywhere. Type the six numbers into the 2×3 grid and the gaussian elimination calculator agrees instantly. Determinant -3, solution [[2],[1]] - read x = 2 first and y = 1 second.

The Gauss-Jordan finish. One more move, $R_1 \leftarrow R_1 + R_2$, gives $\left[\begin{array}{cc|c} 1 & 0 & 2 \\ 0 & 1 & 1 \end{array}\right]$. Now the last column is the answer, no back-substitution needed. That extra sweep is the whole difference between a gaussian elimination calculator and a gauss jordan elimination calculator. One stops at echelon form; the other never stops sweeping.

Three equations, three unknowns - the size where a gaussian elimination calculator really earns its keep. Watch the gaussian elimination calculator's three moves clear one column after another:

x - 2y + 3z = 4, -x + 3y = 0, 2x - 5y + 5z = 6.

1. Write the augmented matrix: $$$$

-1 & 3 & 0 & 0 \ 2 & -5 & 5 & 6 \end{array}\right]$$ 2. $R_2 \leftarrow R_2 + R_1$: row 2 becomes (0, 1, 3 | 4). The x-column is dead below the pivot. 3. $R_3 \leftarrow R_3 - 2R_1$: row 3 becomes (0, -1, -1 | -2). 4. $R_3 \leftarrow R_3 + R_2$: row 3 becomes (0, 0, 2 | 2). 5. Row 3 reads 2z = 2, so z = 1. 6. Row 2 reads y + 3z = 4, so y = 1. 7. Row 1 reads x - 2y + 3z = 4, so x = 3.

Seven lines - that is a gaussian elimination calculator working one column per move.

Verify in all three original equations: 3 - 2 + 3 = 4, -3 + 3 = 0, 6 - 5 + 5 = 6. The solution is (3, 1, 1) - exactly what a gaussian elimination calculator returns for this system. The 3x3 determinant is 2, nonzero, so the unique answer was guaranteed before any arithmetic began. Switch the form to 3×3 and type the twelve entries - the gaussian elimination calculator returns det = 2 and the vector [[3],[1],[1]], the exact procedure walked above.

Now work backwards, from a dead row to the whole solution set - the case every gaussian elimination calculator must explain. Take 2x + 6y = 10 and x + 3y = 5, a pair lifted from a 1904 college algebra text.

1. Write the augmented matrix: $\left[\begin{array}{cc|c} 2 & 6 & 10 \\ 1 & 3 & 5 \end{array}\right]$
2. Eliminate, $R_1 \leftarrow R_1 - 2R_2$: $\left[\begin{array}{cc|c} 0 & 0 & 0 \\ 1 & 3 & 5 \end{array}\right]$
3. The top row reads 0 = 0. Both equations describe the same line, x + 3y = 5.
4. Read the dead row backwards: every point on that line solves the system - (5, 0), (2, 1), (-1, 2), and infinitely many more. The gaussian elimination calculator flagged it with det = 0.

The determinant saw it coming: 2(3) - 6(1) = 0. The gaussian elimination calculator prints that 0 in its first row, and a zero row of zeros means infinitely many solutions.

Change one constant and the story flips. Take 2x + y = 1 with 4x + 2y = 8: the same move $R_2 - 2R_1$ leaves 0 = 6. A false statement means no solution at all - and the gaussian elimination calculator shows det = 0 here too. The two lines run parallel forever.

Type 2, 6, 10, 1, 3, 5 into the gaussian elimination calculator. The determinant row prints 0 and the solution row refuses to print one vector - correctly. The gaussian elimination calculator is not failing; it is reporting that this system has no single answer for any gaussian elimination calculator to give.

Four slips the gaussian elimination calculator catches every day:

1. Columns out of order. x-coefficients go in the first column, y-coefficients in the second. Line both equations up as $ax + by = c$ before typing, or the gaussian elimination calculator solves a different system.
2. Doing the operation to half a row. $R_2 - 2R_1$ must hit every entry of row 2, the constant included. Half a row is the classic silent error the gaussian elimination calculator never makes.
3. Freezing at a 0 pivot. $\left[\begin{array}{cc|c} 0 & 1 & 2 \\ 1 & -1 & 3 \end{array}\right]$ has no leading entry in row 1. Swap the rows first - after the swap the gaussian elimination calculator solves it as (5, 2). Never divide by the 0.
4. Calling det = 0 a bug. The determinant row prints 0 exactly when no unique solution exists. Example 3 shows how the gaussian elimination calculator separates infinitely many solutions from none.

Type carefully and the gaussian elimination calculator does the rest.