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- 3 x 3 Systems Solver
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.
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examples
example 1:ex 1:
Solve by using Gaussian elimination:
$$
\begin{aligned}
x + 2y - z & = 2 \\[2ex]
x - y + 2z & = 5 \\[2ex]
2x + 2y + 2z & = 12
\end{aligned}
$$
example 2:ex 2:
Solve by using Cramer's rule
$$
\begin{aligned}
-x + \frac{2}{3}y - 2z & = 2 \\[2ex]
5x + 7y - 5z & = 6 \\[2ex]
\frac{1}{4}x + y - \frac{1}{2}z & = 2
\end{aligned}
$$
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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