Economics at Zero Marginal Cost: How to solve Simultaneous Equations by Cramer’s Rule (2 equations & 2 Variables)

Simultaneous Equations of n variables and n equations:

$\\a_{11}x_1+a_{12}x_2+a_{13}x_3+...+a_{1n}x_n=d_1\ \\a_{21}x_1+a_{22}x_2+a_{23}x_3+...+a_{2n}x_n=d_2\\\ ..............................................................\\ ..............................................................\\ .............................................................. \\a_{n1}x_1+a_{n2}x_2+a_{n3}x_3+...+a_{nn}x_n=d_n\\\\ where,\\ n=1,2,....,n\\ d_1,d_1,....,d_1\ are\ constant\ terms$

Re-writing the above set of simultaneous equations in matrix form, we have

$\begin{bmatrix} a_{11}\ a_{12}\ .\ . ... a_{1n} \\ a_{21}\ a_{22}\ .\ . ... a_{1n} \\ .................. . ... .. \\ .................. . ... .. \\ a_{n1}\ a_{n2}\ .\ . ... a_{nn} \\ \end{bmatrix} \begin{bmatrix} x_{1}\\ x_{2}\\ .\\ .\\ x_{n} \end{bmatrix} = \begin{bmatrix} d_{1}\\ d_{2}\\ .\\ .\\ d_{n} \end{bmatrix}$

Let,

$A=\begin{bmatrix} a_{11}\ a_{12}\ .\ . ... a_{1n} \\ a_{21}\ a_{22}\ .\ . ... a_{1n} \\ .................. . ... .. \\ .................. . ... .. \\ a_{n1}\ a_{n2}\ .\ . ... a_{nn} \\ \end{bmatrix}, X=\begin{bmatrix} x_{1}\\ x_{2}\\ .\\ .\\ x_{n} \end{bmatrix} and\ d= \begin{bmatrix} d_{1}\\ d_{2}\\ .\\ .\\ d_{n} \end{bmatrix}$

Cramer's Rule:

Steps:

First find the determinant of A, i.e.,|A|

$\Rightarrow$ In order to find the value of the 1st variable $x_{1}$ we need to replace the elements of the 1st column in matrix A by the constant terms $d_{1},\ d_{2},\ ......,\ d_{n}$ and name it as matrix $A_{1}$ .

Next, find the determinant of $A_{1}$ i.e., $\left |A_{1} \right |$ and then divide it by |A| to obtain the value of $x_{1}$ .

i.e.,

$x_{1}=\frac{\left |A_{1} \right |}{\left |A \right |}$

$\Rightarrow$ In order to find the value of the 2nd variable $x_{2}$ we need to replace the elements of the 2nd column in matrix A by the constant terms $d_{1},\ d_{2},\ ......,\ d_{n}$ and name it as matrix $A_{2}$ .

Next, find the determinant of $A_{2}$ i.e., $\left |A_{2} \right |$ and then divide it by |A| to obtain the value of $x_{2}$ .

i.e.,

$x_{2}=\frac{\left |A_{2} \right |}{\left |A \right |}$

$\Rightarrow$ In order to find the value of the 3rd variable $x_{3}$ we need to replace the elements of the 3rd column in matrix A by the constant terms $d_{1},\ d_{2},\ ......,\ d_{n}$ and name it as matrix $A_{3}$ .

Next, find the determinant of $A_{3}$ i.e., $\left |A_{3} \right |$ and then divide it by |A| to obtain the value of $x_{3}$ .

i.e.,

$x_{3}=\frac{\left |A_{3} \right |}{\left |A \right |}$

$\Rightarrow$ In order to find the value of the nth variable $x_{n}$ we need to replace the elements of the nth column in matrix A by the constant terms $d_{1},\ d_{2},\ ......,\ d_{n}$ and name it as matrix $A_{n}$ .

Next, find the determinant of $A_{n}$ i.e., $\left |A_{n} \right |$ and then divide it by |A| to obtain the value of $x_{n}$ .

i.e.,

$x_{n}=\frac{\left |A_{n} \right |}{\left |A \right |}$

1. Solve the following equations by Cramer’s rule:

$\\\3x_{1}-2x_{2}=6\\ 2x_{1}+x_{2}=11$

Solution:

Given: System of simultaneous equations

$\\\3x_{1}-2x_{2}=6\\ 2x_{1}+x_{2}=11$

To find: The values of $x_{1}$ and $x_{2}$ .

Re-writing the equations in matrix form:

$\begin{bmatrix} 3\ \ -2 \\ 2\ \ \ \ \1 \end{bmatrix}\begin{bmatrix} x_{1}\\ x_{2} \end{bmatrix}=\begin{bmatrix} 6\\ 11 \end{bmatrix}$

Let

$A=\begin{bmatrix} 3\ \ -2 \\ 2\ \ \ \ \1 \end{bmatrix},\ X=\begin{bmatrix} x_{1}\\ x_{2} \end{bmatrix}\ and\ d =\begin{bmatrix} 6\\ 11 \end{bmatrix}$

Now,

Next,

$A_{1}=\begin{bmatrix} 6\ \ -2 \\ 11\ \ \ \ \1 \end{bmatrix}\ \ \ \ \ \ A_{2}=\begin{bmatrix} 3\ \ \ \ 6 \\ 2\ \ \11 \end{bmatrix}$

$\\\left |A_{1} \right |=6\times 1-11\times(-2)\\ \left |A_{1} \right |=6+22\\ \left |A_{1} \right |=28\\\ \\\left |A_{2} \right |=3\times 11-2\times6\\ \left |A_{2} \right |=33-12\\ \left |A_{2} \right |=21\\$