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Cramer's Rule Calculator
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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

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

1. ### Solve the system of linear equations by cramer's method:x + y + z = 72x + 3y + 2z = 174x + 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

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

2. ### Solve the system of linear equations by cramer's method:4x – 4y + 5z = 224x – 6y + 3z = 210x – 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