Introduction to Cramer’s Rule Calculator:
Cramer’s rule calculator is an online tool that helps you to find the system of linear equations with the help of the determinant method. Our tool evaluates the various types of linear equations in different order like 2 by 2, 3 by 3, and 4 by 4.
Although the Cramer rule is an easy-to-understand method, it is only for 2 by 2 orders. When you go for a large system of linear equation solutions you need a cramer’s method calculator that helps you to get the solution quickly and easily.
What is Cramer's rule?
Cramer's Rule is a theorem that provides an easy process to solve a system of linear equations with the same number of equations as variables, with the help of the determinant method. It states that the solution xi for each variable xi in the system can be expressed as a ratio of two determinants.
What Is Cramer's Rule Formula?
Cramer rule can be used for the system of linear equations in n by n order. The Cramer rule formula used by the cramer’s rule calculator is as follows,
$$ \begin{matrix} a_{11}x_1 & +\; a_{12}x_2 & +\; … & +\; a_{1n}x_n & \;=\; b_1 \\ a_{21}x_1 & +\; a_{22}x_2 & + … & +\; a_{2n}x_n & \;=\; b_2 \\ \vdots \\ a_{n1}x_1 & +\; a_{n2}x_2 & +\; … & +\; a_{nn}x_n & \;=\; b_n \\ \end{matrix} $$
Such as:
a11,a22,….,ann is the coefficient of x1,x2,….,xnn.
b1,.., bn is the constant in this equation
$$ x \;=\; \frac{det_x}{det},\; y \;=\; \frac{det_y}{det},\; z \;=\; \frac{det_z}{det} $$
Det x is the determinant of the x coefficient
Det y is the determinant of the y coefficient
Det z is the determinant of the z coefficient
How To Calculate Cramer's Rule?
To calculate the system of linear equations using Cramer's Rule step-by-step. Let's see the working method of the cramers rule calculator that will help you to understand how to find the value of variables using determinants.
Step 1:
Identify the System of Equations whether it has 2 by 2 order or 3 by 3 order.
Step 2:
Construct the coefficient matrix A where constants are not included. As per the coefficient make the x matrix,y matrix, and so on, these matrix constants are involved.
$$ \left[ \begin{matrix} A_1 & B_1 \\ A_2 & B_2 \\ \end{matrix} \right] \;=\; \left[ \begin{matrix} C_1 \\ C_2 \\ \end{matrix} \right] $$
X-Matrix:
$$ \left[ \begin{matrix} C_1 & B_1 \\ C_2 & B_2 \\ \end{matrix} \right] $$
Y-matrix:
$$ \left[ \begin{matrix} A_1 & C_1 \\ A_2 & C_2 \\ \end{matrix} \right] $$
Step 3:
Apply the Cramer rule, in which you find the determinant of matrix A and Dx, Dy, and so on as per the given order of the system of equations.
$$ ।D\;। (main\; matrix\; A),\; ।Dx\; ।,\; ।Dy\; ।,\; ।Dz\;।\; and\; ….,\;।Dn\;। $$
Step 4:
To find the value of a given variable from the system of equations divide the determinant solution of ।Dn।,।Dn।,।Dn। by ।D। such as,
$$ X \;=\; \frac{Dx}{D},\; Y \;=\; \frac{Dy}{D}, z \;=\; \frac{Dz}{D} $$
Solved Example Using Cramer Rule:
Let us see a solved example of the Cramer rule to understand how the cramers rule calculator works:
Example: Solve the following equations using cramer’s rule method.
$$ 2x + 5y + 6z \;=\; 16,\; 3x + y + 2z \;=\; 11,\; 4x + 5y -z \;=\; 5 $$
Solution:
The equations can be expressed as,
$$ 2x + 5y + 6z - 16 \;=\; 0 $$
$$ 3x + y + 2z - 11 \;=\; 0 $$
$$ 4x + 5y - z - 5 \;=\; 0 $$
Use the cramer’s rule to determine the values of x, y, z.
$$ \frac{x}{D_x} \;=\; \frac{-y}{D_y} \;=\; \frac{z}{D_z} \;=\; \frac{-1}{D} $$
$$ D_x \;=\; \left| \begin{matrix} 5 & 6 & -16 \\ 1 & 2 & -11 \\ 5 & -1 & -5 \\ \end{matrix} \right] $$
$$ =\; 5 \times \left[ \begin{matrix} 2 & -11 \\ -1 & -5 \\ \end{matrix} \right] -6 \times \left[ \begin{matrix} 1 & -11 \\ 5 & -5 \\ \end{matrix} \right] -16 \times \left[ \begin{matrix} 1 & 2 \\ 5 & -1 \\ \end{matrix} \right] $$
$$ 5 \times (2 \times (-5) - (-11) \times (-1)) -6 \times (1 \times (-5) - (-11) \times 5) - 16 \times (1 \times (-1) - 2 \times 5 ) $$
$$ =\; 5 \times (-10 - 11) - 6 \times (-5 + 55) - 16 \times (-1 - 10) $$
$$ =\; 5 \times (-21) - 6 \times (50) - 16 \times (-11) $$
$$ =\; -105 - 300 + 176 $$
$$ =\; -229 $$
$$ D_y \;=\; \left| \begin{matrix} 2 & 6 & -16 \\ 3 & 2 & -11 \\ 4 & -1 & -5 \\ \end{matrix} \right| $$
$$ =\; 2 \times \left| \begin{matrix} 2 & -11 \\ -1 & -5 \\ \end{matrix} \right| -6 \times \left| \begin{matrix} 3 & -11 \\ 4 & -5 \\ \end{matrix} \right| -16 \times \left| \begin{matrix} 3 & 2 \\ 4 & -1 \\ \end{matrix} \right| $$
$$ =\; 2 \times (2 \times (-5) - (-11) \times (-1)) - 6 \times (3 \times (-5) - (-11) \times 4) -16 \times (3 \times (-1) - 2 \times 4) $$
$$ =\; 2 \times (-10 - 11) -6 \times (-15 + 44) - 16 \times (-3 - 8) $$
$$ 2 \times (-21) - 6 \times (29) -16 \times (-11) $$
$$ =\; -40 $$
$$ D_z \;=\; \left| \begin{matrix} 2 & 5 & -16 \\ 3 & 1 & -11 \\ 4 & 5 & -5 \\ \end{matrix} \right| $$
$$ =\; 2 \times \left| \begin{matrix} 1 & -11 \\ 5 & -5 \\ \end{matrix} \right| -5 \times \left| \begin{matrix} 3 & -11 \\ 4 & -5 \\ \end{matrix} \right| -16 \times \left| \begin{matrix} 3 & 1 \\ 4 & 5 \\ \end{matrix} \right| $$
$$ 2 \times (1 \times (-5) - (-11) \times 5) -5 \times (3 \times (-5) - (-11) \times 4) - 16 \times (3 \times 5 -1 \times 4) $$
$$ =\; 2 \times (-5 + 55) -5 \times (-15 + 44) -16 \times (15 - 4) $$
$$ =\; 2 \times (50) - 5 \times (29) - 16 \times (11) $$
$$ =\; 100 - 145 - 176 $$
$$ =\; -221 $$
$$ D \;=\; \left| \begin{matrix} 2 & 5 & 6 \\ 3 & 1 & 2 \\ 4 & 5 & -1 \\ \end{matrix} \right| $$
$$ =\; 2 \times \left| \begin{matrix} 1 & 2 \\ 5 & -1 \\ \end{matrix} \right| -5 \times \left| \begin{matrix} 3 & 2 \\ 4 & -1 \\ \end{matrix} \right| +6 \times \left| \begin{matrix} 3 & 1 \\ 4 & 5 \\ \end{matrix} \right| $$
$$ =\; 2 \times (1 \times (-1) -2 \times 5) - 5 \times (3 \times (-1) -2 \times 4) + 6 \times (3 \times 5 - 1 \times 4) $$
$$ =\; 2 \times (-1 - 10) -5 \times (-3 - 8) + 6 \times (15 - 4) $$
$$ =\; 2 \times (-11) -5 \times (-11) + 6 \times (11) $$
$$ =\; -22 + 55 + 66 $$
$$ 99 $$
Add all the values to get the values of x , y and z.
$$ \frac{x}{D_x} \;=\; \frac{-y}{D_y} \;=\; \frac{z}{D_z} \;=\; \frac{-1}{D} $$
$$ \frac{x}{-229} \;=\; \frac{-y}{-40} \;=\; \frac{z}{-221} \;=\; \frac{-1}{99} $$
$$ \frac{x}{-229} \;=\; \frac{-1}{99}. \frac{-y}{-40} \;=\; \frac{-1}{99}, \frac{z}{-221} \;=\; \frac{-1}{99} $$
$$ x \;=\; \frac{229}{99},\; y \;=\; \frac{-40}{99},\; z \;=\; \frac{221}{99} $$
$$ x \;=\; \frac{229}{99},\; y \;=\; -\frac{40}{99}, z \;=\; \frac{221}{99} $$
How To Use The Cramer's Method Calculator?
Cramer's rule 3x3 calculator has a user-friendly interface so you just need to enter your problem in this calculator to get a solution in an easy method. Follow our guidelines before using it . These guidelines are:
- Select the matrix size (2 by 2 or maybe 3 by 3) from the given list for finding the determinant of matrix using cramer rule.
- Enter the value of the system of linear equations in the input box of cramers rule calculator.
- Review the given equation value before hitting the calculate button to start the evaluation process in the cramers law calculator
- Click the “Calculate” button to get the solution of your given linear equation problem.
- If you want to try out our professional tool for the first time then you must try out the load example to learn more about it.
- Click on the “Recalculate” button to get a new page for finding more example solutions of Cramer rule problems.
Final Result Of The Cramer's Rule Calculator:
Cramer’s method calculator gives you the solution to a given linear equation question when you add the input into it. It may be included as:
Result option:
When you click on the result option it gives you a solution to the Cramer rule problem.
Possible steps:
It provides you with a solution to the Cramer rule problem where all the calculations are mentioned in steps.
Advantages Of Using The Cramers Law Calculator:
Cramer rule Calculator provides a ton of advantages that you get when you calculate the system of linear equation problems and provides solutions immediately. These advantages are:
- Cramer’s rule 3x3 calculator is a free tool that enables you to evaluate the Cramer rule problem with a solution.
- It is a manageable tool that can solve different orders of equations to find the value of a variable using the Cramer rule.
- The Cramer’s rule calculator saves the time that you consume on the calculation of the linear equation solution with the help of the Cramer rule in a couple of minutes.
- Our tool helps you to get a stronghold on the Cramer rule method when you use it for practice.
- Cramer law Calculator is a reliable tool that provides accurate solutions when you use it to calculate the Cramer rule problems without any error.
- Cramer rule Calculator provides the solution without a sign-in condition so you can use it anywhere through the internet.
