3x3 system of equations solver

This calculator solves system of three equations with three unknowns (3x3 system). The calculator will use the Gaussian elimination or Cramer's rule to generate a step by step explanation.

3x3 System of equations solver

two solving methods + detailed steps
help ↓↓ examples ↓↓ tutorial ↓↓

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Enter system of equations (empty fields will be replaced with zeros)

Choose computation method:

Solve using Gaussian elimination method (default)

Solve using Cramer's rule

Settings:

Find approximate solution

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Examples

ex 1:

Solve using Gaussian elimination:

x + 2 y - z x - y + 2 z 2 x + 2 y + 2 z = 2 = 5 = 12

ex 2:

Solve using Cramer's rule

- x + \frac{2}{3} y - 2 z 5 x + 7 y - 5 z \frac{1}{4} x + y - \frac{1}{2} z = 2 = 6 = 2

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About Cramer's rule

This calculator uses Cramer's rule to solve systems of three equations with three unknowns. The Cramer's rule can be stated as follows:

Given the system:

a_{1} x + b_{1} y + c_{1} z = d_{1} a_{2} x + b_{2} y + c_{2} z = d_{2} a_{3} x + b_{3} y + c_{3} z = d_{3}

with

D = a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3} \neq = 0

D_{x} = d_{1} d_{2} d_{3} b_{1} b_{2} b_{3} c_{1} c_{2} c_{3}

D_{y} = a_{1} a_{2} a_{3} d_{1} d_{2} d_{3} c_{1} c_{2} c_{3}

D_{z} = a_{1} a_{2} a_{3} b_{1} b_{2} b_{3} d_{1} d_{2} d_{3}

then the solution of this system is:

x = \frac{D _{x}}{D}

y = \frac{D _{y}}{D}

z = \frac{D _{z}}{D}

Example: Solve the system of equations using Cramer's rule

4 x + 5 y - 2 z = 7 x - y + 2 z = 3 x + y + 4 z = - 14 4228

Solution: First we compute $D, D_{x}, D_{y}$ and $D_{z}$ .

D = 473 5 - 1 1 - 2 24 = - 16 + 30 - 14 - 6 - 8 - 140 = - 154 D_{x} = - 14 4228 5 - 1 1 - 2 24 = 56 + 280 - 84 - 56 + 28 - 840 = - 616 D_{y} = 473 - 14 4228 - 2 24 = 672 - 84 - 392 + 252 - 224 + 392 = 616 D_{Z} = 473 5 - 1 1 - 14 4228 = - 112 + 630 - 98 - 42 - 168 - 980 = - 770

Therefore,

x = \frac{D _{x}}{D} = \frac{- 616}{- 154} = 4 y = \frac{D _{y}}{D} = \frac{616}{- 154} = - 4 z = \frac{D _{z}}{D} = \frac{- 770}{- 154} = 5

Note: You can check the solution using above calculator

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