What is back substitution?

Back substitution solves a triangular system from its last equation upward. The bottom row holds a single unknown, so it is solved first. That value goes into the row above, which then has one unknown left, and so on to the top. If the last row reads 2x₄ = 10, then x₄ = 5, and the row above uses that 5.

How do you find the rank of a matrix?

To find the rank of a matrix, row-reduce it until a staircase of leading entries appears, then count the rows that are not all zeros. A row that turns into zeros was a mix of the others and adds nothing new. A 3×3 grid whose third row is the sum of the first two has rank 2.

Why do I get the wrong sign in row reduction?

Wrong signs in row reduction usually come from subtracting a negative entry. If row 1 holds −3 and the move is R₂ − 2·R₁, you subtract 2 × (−3) = −6, which means adding 6. Write each product in brackets with its sign before subtracting, and the slip disappears.

When do students learn Gaussian elimination?

In many school systems, Gaussian elimination arrives with a first university course in linear algebra, though some secondary schools meet it through 3×3 systems. Here, its levels are aimed at 18-year-olds. Anyone at ease with negative numbers and simple equations can start on the first level.

Is this row reduction game free, and can a class play it?

Yes, it is free in any browser on laptops, tablets or phones, with nothing to install. For row reduction practice, give each team a 3×3 system and have them call out every new entry as they eliminate. The first team to reach a clean staircase explains its moves to the room. The schools plan, priced per student, adds a teacher dashboard showing who is getting quicker.

Math practice · reimagined

Gaussian Elimination Game

Here Gaussian elimination comes one entry at a time: the number a row move leaves behind, an unknown read off a staircase, or a whole grid’s rank. Your rival reads the same matrix, so the cleaner your arithmetic, the sooner your fighter swings.

Ages 4–99+ · no download · plays in your browser · 23 languages

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What is Gaussian elimination?

Gaussian elimination is a method for solving a system of linear equations by tidying its matrix, the grid of coefficients, one row at a time. You may swap two rows, scale a row, or subtract a multiple of one row from another. Step by step, zeros pile up below a staircase, and the answers can then be read from the bottom.

Every level of Gaussian elimination

Example: R₂ − 5·R₁ on [ 1 2 4 8 ; 5 4 6 7 ; 5 4 5 8 ]

The question asks for a₂₂, the new entry in row 2, column 2, after R₂ − 5·R₁.
In that column, row 2 holds 4 and row 1 holds 2.
Take five copies of the row 1 entry away: 4 − 5 × 2 = 4 − 10 = −6.
The new entry is −6, so you type it with the minus sign.

Learn it step by step

Gaussian elimination tidies a matrix one row at a time. The move it uses most is: take a row and subtract a number of copies of another row from it. Written R₂ − 3·R₁, it means: new row 2 = row 2 minus 3 times row 1.

You do it entry by entry, column by column. To find the new entry in row 2, column 3, take the old row 2 entry in column 3 and subtract 3 times the row 1 entry in column 3. If they are 7 and 4, the new entry is 7 − 3 × 4 = 7 − 12 = −5. The k is picked so the first entry of the row becomes 0 — that's the point of the move — but you only need the one entry the question names.

Find the two entries in the column asked for, multiply the top one by k, subtract, and type the new entry.

Tips and common mistakes

Read a₃₂ like a seat in a cinema: row first, then seat. Row 3, column 2, never the other way round.
A common slip is changing both rows. R₂ − 5·R₁ rewrites row 2 only; row 1 stays exactly as it was and just lends its entries.
Write a small system on paper, such as x + y = 5 and x − y = 1, and race to eliminate x before solving for y.

How to play

Each round puts a matrix on screen with one question beside it: a row move and the entry it changes, an unknown from a triangular system, or a rank. Type the number on the keypad before your rival and Garlic lands the hit, harder the faster you were. A wrong answer stuns you for a moment.

— Quick answers —

Gaussian elimination: the short FAQ

What is back substitution?: Back substitution solves a triangular system from its last equation upward. The bottom row holds a single unknown, so it is solved first. That value goes into the row above, which then has one unknown left, and so on to the top. If the last row reads 2x₄ = 10, then x₄ = 5, and the row above uses that 5.
How do you find the rank of a matrix?: To find the rank of a matrix, row-reduce it until a staircase of leading entries appears, then count the rows that are not all zeros. A row that turns into zeros was a mix of the others and adds nothing new. A 3×3 grid whose third row is the sum of the first two has rank 2.
Why do I get the wrong sign in row reduction?: Wrong signs in row reduction usually come from subtracting a negative entry. If row 1 holds −3 and the move is R₂ − 2·R₁, you subtract 2 × (−3) = −6, which means adding 6. Write each product in brackets with its sign before subtracting, and the slip disappears.
When do students learn Gaussian elimination?: In many school systems, Gaussian elimination arrives with a first university course in linear algebra, though some secondary schools meet it through 3×3 systems. Here, its levels are aimed at 18-year-olds. Anyone at ease with negative numbers and simple equations can start on the first level.
Is this row reduction game free, and can a class play it?: Yes, it is free in any browser on laptops, tablets or phones, with nothing to install. For row reduction practice, give each team a 3×3 system and have them call out every new entry as they eliminate. The first team to reach a clean staircase explains its moves to the room. The schools plan, priced per student, adds a teacher dashboard showing who is getting quicker.

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