Question 15.19: Find x1 and x2 using Cramer’s rule from the following set of......
Find x_{1} and x_{2} using Cramer’s rule from the following set of simultaneous equations
5 x_{1} + 0.4 x_{2} = 12
3 x_{1} + 3 x_{2} = 21
These simultaneous equations can be represented in matrix format as
Ax = \begin{bmatrix} 5& 0.4 \\ 3& 3 \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2} \end{bmatrix} = \begin{bmatrix}12\\21 \end{bmatrix} = b
Using Cramer’s rule to find x_{1} by substituting the vector b of constants for column 1 in matrix A gives
x_{1} = \frac{\left|A_{1}\right| }{\left| A\right| } = \frac{\begin{vmatrix} 12& 0.4 \\ 21& 3 \end{vmatrix}}{\begin{vmatrix}5 &0.4 \\ 3 & 3 \end{vmatrix} } = \frac{36 − 8.4}{15 − 1.2} = \frac{27.6}{13.8} = 2
In a similar fashion, by substituting vector b for column 2 in matrix A we get
x_{2} = \frac{\left|A_{2}\right| }{\left| A\right| } = \frac{\begin{vmatrix} 5& 12 \\ 3& 21 \end{vmatrix}}{\begin{vmatrix}5 &0.4 \\ 3 & 3 \end{vmatrix} } = \frac{105 – 36}{15 − 1.2} = \frac{69}{13.8} = 5