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Gaussian Elimination

Each equation is a plane, and solving the system is finding where all three meet. Row operations swap one plane for another — but never move the meeting point.

One solution · step 1 of 41

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System 1: 2x + y − z = 8; −3x − y + 2z = −11; −2x + y + 2z = −3. Each equation is a plane in 3D — green, red and blue. A solution is a point on all three at once. Row operations change the planes but never that common point.

R₁	2	1	−1	8
R₂	−3	−1	2	−11
R₃	−2	1	2	−3

2x + y − z = 8
−3x − y + 2z = −11
−2x + y + 2z = −3

\left[\begin{array}{rrr|r} \textcolor{#4ade80}{2} & \textcolor{#4ade80}{1} & \textcolor{#4ade80}{-1} & \textcolor{#4ade80}{8} \\ \textcolor{#f87171}{-3} & \textcolor{#f87171}{-1} & \textcolor{#f87171}{2} & \textcolor{#f87171}{-11} \\ \textcolor{#60a5fa}{-2} & \textcolor{#60a5fa}{1} & \textcolor{#60a5fa}{2} & \textcolor{#60a5fa}{-3} \end{array}\right]

Gaussian Elimination

Each equation is a plane, and solving the system is finding where all three meet. Row operations swap one plane for another — but never move the meeting point.

One solution · step 1 of 41

Speed

R₁	2	1	−1	8
R₂	−3	−1	2	−11
R₃	−2	1	2	−3

2x + y − z = 8
−3x − y + 2z = −11
−2x + y + 2z = −3

\left[\begin{array}{rrr|r} \textcolor{#4ade80}{2} & \textcolor{#4ade80}{1} & \textcolor{#4ade80}{-1} & \textcolor{#4ade80}{8} \\ \textcolor{#f87171}{-3} & \textcolor{#f87171}{-1} & \textcolor{#f87171}{2} & \textcolor{#f87171}{-11} \\ \textcolor{#60a5fa}{-2} & \textcolor{#60a5fa}{1} & \textcolor{#60a5fa}{2} & \textcolor{#60a5fa}{-3} \end{array}\right]

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Gaussian Elimination

See also

Gaussian Elimination

See also