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Calculators :: System of equations solvers :: System 3x3

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 calculator

You can enter integers (10), decimal numbers(10.12) or fractions(10/3).
Important: The form will NOT let you enter wrong characters (like *, (, ), x, p,...)

Solve by using Gaussian elimination method (default)
Solve by using Cramer's rule
Show me the solution without steps

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:

$$ \begin{aligned} a_1x + b_1y + c_1z = d_1 \\ a_2x + b_2y + c_2z = d_2 \\ a_3x + b_3y + c_3z = d_3 \end{aligned} $$

with

$$ D = \left|\begin{array}{ccc} a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2 \\ a_3 & b_3 & c_3 \\ \end{array}\right| \ne 0 $$ $$ D_x = \left|\begin{array}{ccc} d_1 & b_1 & c_1 \\ d_2 & b_2 & c_2 \\ d_3 & b_3 & c_3 \\ \end{array}\right| $$ $$ D_y = \left|\begin{array}{ccc} a_1 & d_1 & c_1 \\ a_2 & d_2 & c_2 \\ a_3 & d_3 & c_3 \\ \end{array}\right| $$ $$ D_z = \left|\begin{array}{ccc} a_1 & b_1 & d_1 \\ a_2 & b_2 & d_2 \\ a_3 & b_3 & d_3 \\ \end{array}\right| $$

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

$$ \begin{aligned} 4x + 5y -2z= & -14 \\ 7x - ~y +2z= & 42 \\ 3x + ~y + 4z= & 28 \\ \end{aligned} $$

Solution: First we compute $ D,~ D_x,~ D_y $ and $ D_z $.

$$ \begin{aligned} & D~~ = \left|\begin{array}{ccc} {\color{blue}{4}} & {\color{red}{~5}} & {\color{green}{-2}} \\ {\color{blue}{7}} & {\color{red}{-1}} & {\color{green}{~2}} \\ {\color{blue}{3}} & {\color{red}{~1}} & {\color{green}{~4}} \end{array}\right| = -16 + 30 - 14 - 6 - 8 - 140 = -154\\ & D_x = \left|\begin{array}{ccc} -14 & {\color{red}{~5}} & {\color{green}{-2}} \\ ~42 & {\color{red}{-1}} & {\color{green}{~2}} \\ ~28 & {\color{red}{1}} & {\color{green}{~4}} \end{array}\right| = 56 + 280 - 84 - 56 + 28 - 840 = -616\\ & D_y = \left|\begin{array}{ccc} {\color{blue}{4}} & -14 & {\color{green}{-2}} \\ {\color{blue}{7}} & ~42 & {\color{green}{~2}} \\ {\color{blue}{3}} & ~28 & {\color{green}{~4}} \end{array}\right| = 672 - 84 - 392 + 252 - 224 + 392 = 616\\ & D_Z = \left|\begin{array}{ccc} {\color{blue}{4}} & {\color{red}{~5}} & -14 \\ {\color{blue}{7}} & {\color{red}{-1}} & ~42 \\ {\color{blue}{3}} & {\color{red}{~1}} & ~28 \end{array}\right| = -112 + 630 - 98 - 42 - 168 - 980 = -770\\ \end{aligned} $$

Therefore,

$$ \begin{aligned} & x = \frac{D_x}{D} = \frac{-616}{-154} = 4 \\ & y = \frac{D_y}{D} = \frac{ 616}{-154} = -4 \\ & z = \frac{D_z}{D} = \frac{-770}{-154} = 5 \end{aligned} $$

Note: You can check the solution using above calculator

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