# Cramer's Rule Calculator

## Solve the system of linear equations using Cramer's rule step by step

This calculator will solve the system of linear equations of any kind, with steps shown, using Cramer's rule.

Related calculators: System of Equations Calculator, System of Linear Equations Calculator

The Cramer's Rule Calculator is an online tool that implements one of the methods for solving systems of linear equations. This method, named after the Swiss mathematician Gabriel Cramer, is known as Cramer's Rule. It is a mathematical formula that provides a systematic approach for obtaining a solution.

## What Is the Process for Calculating Cramer's Rule?

### Step 1

To apply Cramer's Rule, you must begin with a system of equations that is characterized by an equal number of equations and variables. If this criterion isn't met, Cramer's Rule won't be applicable.

### Step 2

Translate your system of equations into matrix form, denoted as $$$Ax=b$$$. Here, matrix $$$A$$$ is an $$$n\times n$$$ matrix encompassing the coefficients of the variables, where $$$A_{ij}$$$ is the coefficient paired with the $$$j$$$-th variable in the $$$i$$$-th equation. Meanwhile, $$$b$$$ symbolizes a vector (of size $$$n$$$) that assembles the right-hand side of each equation.

### Step 3

Determine the determinant of the matrix $$$A$$$, denoted by $$$\left|A\right|$$$. If $$$\left|A\right|$$$ equals zero, the system doesn't have a unique solution, thus rendering Cramer's Rule ineffective.

### Step 4

Construct the associated matrix $$$A_j$$$, which is identical to matrix $$$A$$$ but with the $$$j$$$-th column of $$$A$$$ substituted by $$$b$$$. Afterwards, compute its determinant.

### Step 5

If $$$\left|A\right|$$$ is not equal to zero, there exists a unique solution. The elements of the solution, denoted as $$$x_j$$$ for $$$j=\overline{1,2,..n}$$$, can then be calculated as follows:$$$x_j=\frac{\left|A_j\right|}{\left|A\right|}$$$.

## Using the Cramer's Rule Calculator

Our Cramer's Rule Calculator is simple and intuitive. Follow these steps:

### Step One

Input the coefficients of your system of equations into the designated fields on the calculator.

### Step Two

Click the "Calculate" button.

### Step Three

The solutions will appear in the "Results" box immediately.

## Cramer's Rule formula

Remember, the calculator uses Cramer's Rule formula to solve the system of equations, where, for each variable, the system determinant is replaced with a determinant formed by replacing the coefficients of the respective variable with constants, and then divided by the determinant of coefficients:

For a system of two equations:

$$\begin{cases}a_1x+b_1y=e_1\\a_2x+b_2y=e_2\end{cases}$$The solutions can be found using Cramer's Rule as follows:

$$\Delta=a_1b_2-a_2b_1$$ $$\Delta_x=e_1b_2-e_2b_1$$ $$\Delta_y=a_1e_2-a_2e_1$$

$$x=\frac{\Delta_x}{\Delta}$$ $$y=\frac{\Delta_y}{\Delta}$$

Here, $$$\Delta$$$ is the determinant of the coefficient matrix, $$$\Delta_x$$$ and $$$\Delta_y$$$ are determinants formed by replacing the coefficients of $$$x$$$ and $$$y$$$ respectively with the constant terms in the equations.

Similarly, for a system of three equations:

$$\begin{cases}a_1x+b_1y+c_1z=e_1\\a_2x+b_2y+c_2z=e_2\\a_3x+b_3y+c_3z=e_3\end{cases}$$The solutions can be found as:

$$\Delta=a_1(b_2c_3-b_3c_2)-b_1(a_2c_3-a_3c_2)+c_1(a_2b_3-a_3b_2)$$ $$\Delta_x=e_1(b_2c_3-b_3c_2)-b_1(e_2c_3-e_3c_2)+c_1(e_2b_3-e_3b_2)$$ $$\Delta_y=a_1(e_2c_3-e_3c_2)-e_1(a_2c_3-a_3c_2)+c_1(a_2e_3-a_3e_2)$$ $$\Delta_z=a_1(b_2e_3-b_3e_2)-b_1(a_2e_3-a_3e_2)+e_1(a_2b_3-a_3b_2)$$

$$x=\frac{\Delta_x}{\Delta}$$ $$y=\frac{\Delta_y}{\Delta}$$ $$z=\frac{\Delta_z}{\Delta}$$

In each case, the solution for each variable is the ratio of the determinant formed by replacing the coefficients of that variable with the constant terms to the determinant of the coefficients.

## Avoiding Common Mistakes

Cramer's Rule only works with square systems of linear equations where the determinant of the coefficient matrix $$$\Delta$$$ is non-zero. Our calculator will help you avoid the error of using Cramer's Rule for unsuitable systems by giving a prompt if the entered system is not appropriate.

## Advanced Tips

While our calculator is a valuable tool for making calculations easier, it's essential to also understand the underlying theory. Additionally, be aware that, while Cramer's Rule is excellent for small systems, it might not be the most efficient for larger ones due to the computational cost of calculating determinants.

### FAQ

#### When is Cramer's Rule most useful?

It is typically most effective when dealing with small systems of 2 or 3 linear equations.

#### Which systems can't be solved with Cramer's Rule?

Systems that are not square or those with a determinant of zero can't be solved using Cramer's Rule.

#### What should I do if the calculator isn't working?

First, make sure your inputs are correct. If the issue persists, refresh the page or contact our support team.