Show 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. examples Solve by using Gaussian elimination: $$ \begin{aligned} x + 2y - z & = 2 \\[2ex] x - y + 2z & = 5 \\[2ex] 2x + 2y + 2z & = 12 \end{aligned} $$ 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 ruleThis 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
then the solution of this system is:
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 Search our database of more than 200 calculators 227 993 127 solved problems This online calculator will help you to solve a system of linear equations using Cramer's rule. Using this online calculator, you will receive a detailed step-by-step solution to your problem, which will help you understand the algorithm how to solve system of linear equations using Cramer's rule. Calculator Guide The number of equations in the system: Change the names of the variables in the system Fill the system of linear equations: You can input only integer numbers, decimals or fractions in this online calculator (-2.4, 5/7, ...). More in-depth information read at these rules.
How do you solve a system using determinants?How to solve a system of two equations using Cramer's rule.. Evaluate the determinant D, using the coefficients of the variables.. Evaluate the determinant. ... . Evaluate the determinant. ... . Find x and y.. Write the solution as an ordered pair.. Check that the ordered pair is a solution to both original equations.. |
Solve the system of equations using determinants calculator
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