To get the best deal on Tutoring, call 1-855-666-7440 (Toll Free)
Top

Cramer's Rule Calculator
Top
In determinants, the concept of Cramer’s Rule is to solve the simultaneous equations. Any given set of simultaneous equations can be converted to a matrix form as $AX$ = $B$. Then the values of each variable is obtained separately and directly by applying the determinants.

Cramer's Rule: If $AX$ = $B$ is a system of n-linear equations with n-unknowns such that $det(A)\ \neq\ 0$, then the system has a unique solution. 

The solution is,

$x_1$ = $\frac{det(A_{1})}{det(A)}$, $x_2$ = $\frac{det(A_{2})}{det(A)}$, $x_3$ = $\frac{det(A_{3})}{det(A)}$ ............., $x_n$ = $\frac{det(A_{n})}{det(A)}$   

Where $A_j$ is the matrix obtained by replacing the entries in the jth column of $A$ by the entries in the matrix 

$B$ = $\begin{bmatrix} b_{1}\\ b_{2}\\ ...\\ ...\\ b_{n}\\ \end{bmatrix}$

Cramer's Rule Calculator is will help us to find the solution for a given system of linear equations.
 

Steps for Cramer's Rule Calculator

Back to Top
Step 1 :  

Observe the given system of linear equations and write the coefficient matrix(A). Then find det(A).



Step 2 :  

Now write the matrix Ax by substituting the first column of matrix A with constants and find det(Ax). Similarly find det(Ay) and det(Az).



Step 3 :  

Now use cramer's rule:


x = $\frac{det(A_{x})}{det(A)}$


y = $\frac{det(A_{y})}{det(A)}$


z = $\frac{det(A_{z})}{det(A)}$



Problems on Cramer's Rule Calculator

Back to Top
  1. Solve the system of linear equations by cramer's method:

    x + y + z = 7

    2x + 3y + 2z = 17

    4x + 9y + z = 37


    Step 1 :  

    Given equations:


    x + y + z = 7


    2x + 3y + 2z = 17


    4x + 9y + z = 37


    Matrix(A) = $\begin{bmatrix}
    1 & 1 & 1\\
    2 & 3 & 2\\
    4 & 9 & 1
    \end{bmatrix}$


    Det(A) = $\begin{vmatrix}
    1 & 1 & 1\\
    2 & 3 & 2\\
    4 & 9 & 1
    \end{vmatrix}$


    Det(A) = -3



    Step 2 :  

    Matrix(Ax) = $\begin{bmatrix}
    7 & 1 & 1\\
    17 & 3 & 2\\
    37 & 9 & 1
    \end{bmatrix}$


    Det(Ax) = $\begin{vmatrix}
    7 & 1 & 1\\
    17 & 3 & 2\\
    37 & 9 & 1
    \end{vmatrix}$


    Det(Ax) = -6


    Matrix(Ay) = $\begin{bmatrix}
    1 & 7 & 1\\
    2 & 17 & 2\\
    4 & 37 & 1
    \end{bmatrix}$


    Det(Ay) = $\begin{vmatrix}
    1 & 7 & 1\\
    2 & 17 & 2\\
    4 & 37 & 1
    \end{vmatrix}$


    Det(Ay) = -9


    Matrix(Az) = $\begin{bmatrix}
    1 & 1 & 7\\
    2 & 3 & 17\\
    4 & 9 & 37
    \end{bmatrix}$


    Det(Az) = $\begin{vmatrix}
    1 & 1 & 7\\
    2 & 3 & 17\\
    4 & 9 & 37
    \end{vmatrix}$


    Det(Az) = -6



    Step 3 :  

    x = $\frac{-6}{-3}$ = 2


    y = $\frac{-9}{-3}$ = 3


    z = $\frac{-6}{-3}$ = 2



    Answer  :  

    (x, y, z) = (2, 3, 2)



  2. Solve the system of linear equations by cramer's method:

    4x – 4y + 5z = 22
    4x – 6y + 3z = 2
    10x – 3y + z = 14


    Step 1 :  

    Given equations:


    4x – 4y + 5z = 22
    4x – 6y + 3z = 2
    10x – 3y + z = 14


    Matrix(A) = $\begin{bmatrix}
    4 & -4 & 5\\
    4 & -6 & 3\\
    10 & -3 & 1
    \end{bmatrix}$


    Det(A) = $\begin{vmatrix}
    4 & -4 & 5\\
    4 & -6 & 3\\
    10 & -3 & 1
    \end{vmatrix}$


    Det(A) = 148



    Step 2 :  

    Matrix(Ax) = $\begin{bmatrix}
    22 & -4 & 5\\
    2 & -6 & 3\\
    14 & -3 & 1
    \end{bmatrix}$


    Det(Ax) = $\begin{vmatrix}
    22 & -4 & 5\\
    2 & -6 & 3\\
    14 & -3 & 1
    \end{vmatrix}$


    Det(Ax) = 296


    Matrix(Ay) = $\begin{bmatrix}
    4 & 22 & 5\\
    4 & 2 & 3\\
    10 & 14 & 1
    \end{bmatrix}$


    Det(Ay) = $\begin{vmatrix}
    4 & 22 & 5\\
    4 & 2 & 3\\
    10 & 14 & 1
    \end{vmatrix}$


    Det(Ay) = 592


    Matrix(Az) = $\begin{bmatrix}
    4 & -4 & 22\\
    4 & -6 & 2\\
    10 & -3 & 14
    \end{bmatrix}$


    Det(Az) = $\begin{vmatrix}
    4 & -4 & 22\\
    4 & -6 & 2\\
    10 & -3 & 14
    \end{vmatrix}$


    Det(Az) = 888



    Step 3 :  

    x = $\frac{296}{148}$ = 2


    y = $\frac{592}{148}$ = 4


    z = $\frac{888}{148}$ = 6



    Answer  :  

    (x, y, z) = (2, 4, 6)



*AP and SAT are registered trademarks of the College Board.