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

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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=bAx=b. Here, matrix AA is an n×nn\times n matrix encompassing the coefficients of the variables, where AijA_{ij} is the coefficient paired with the jj-th variable in the ii-th equation. Meanwhile, bb symbolizes a vector (of size nn) that assembles the right-hand side of each equation.

  • Step 3

    Determine the determinant of the matrix AA, denoted by A\left|A\right|. If A\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 AjA_j, which is identical to matrix AA but with the jj-th column of AA substituted by bb. Afterwards, compute its determinant.

  • Step 5

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

{a1x+b1y=e1a2x+b2y=e2\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:

Δ=a1b2a2b1\Delta=a_1b_2-a_2b_1 Δx=e1b2e2b1\Delta_x=e_1b_2-e_2b_1 Δy=a1e2a2e1\Delta_y=a_1e_2-a_2e_1

x=ΔxΔx=\frac{\Delta_x}{\Delta} y=ΔyΔy=\frac{\Delta_y}{\Delta}

Here, Δ\Delta is the determinant of the coefficient matrix, Δx\Delta_x and Δy\Delta_y are determinants formed by replacing the coefficients of xx and yy respectively with the constant terms in the equations.

Similarly, for a system of three equations:

{a1x+b1y+c1z=e1a2x+b2y+c2z=e2a3x+b3y+c3z=e3\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:

Δ=a1(b2c3b3c2)b1(a2c3a3c2)+c1(a2b3a3b2)\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) Δx=e1(b2c3b3c2)b1(e2c3e3c2)+c1(e2b3e3b2)\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) Δy=a1(e2c3e3c2)e1(a2c3a3c2)+c1(a2e3a3e2)\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) Δz=a1(b2e3b3e2)b1(a2e3a3e2)+e1(a2b3a3b2)\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=ΔxΔx=\frac{\Delta_x}{\Delta} y=ΔyΔy=\frac{\Delta_y}{\Delta} z=ΔzΔz=\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 Δ\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.