Question

Using Cramer’s Rule, what is the value of x in the system of linear equations below?

Answer 1

Answer:  The required value of x is 1.

Step-by-step explanation:  We are to use the Cramer's rule to find the value of x in the following system of equations :

$x+3y=16~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(i)\\\\3x+y=8~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~(ii)$

By Cramer's rule, we get

$\dfrac{x}{D_1}=\dfrac{y}{D_2}=\dfrac{1}{D},$

where

$D_1=\begin{bmatrix}16 & 3\\ 8 & 1\end{bmatrix}=16\times1-8\times3=16-24=-8,\\\\\\\\D_2=\begin{bmatrix}1 & 16\\ 3 & 8\end{bmatrix}=8\times1-16\times3=8-48=-40,\\\\\\\\D=\begin{bmatrix}1 & 3\\ 3 & 1\end{bmatrix}=1\times1-3\times3=1-9=-8.$

Therefore, we get

$\dfrac{x}{-8}=\dfrac{y}{-40}=\dfrac{1}{-8}\\\\\\\Rightarrow x=\dfrac{-8}{-8},~~~y=\dfrac{-40}{-8}\\\\\\\Rightarrow x=1,~~y=5.$

Thus, the required value of x is 1.

Answer 2

We know that
Cramer's rule works as follows:
x+3y=16
3x+y=8
Then
x=Dx/D
y=Dy/D
where Dx,Dy,D are 2x2 matrices formed from of coefficients and right hand side.
D=
1 3
3 1
=1-9=-8
Dx=matrix D with first column replaced by the vector [16,8]=
16 3
 8  1
=16-24=-8
Dy=matrix D with second column replaced by the vector [16,8]=
1 16
3  8
=8-48=-40

therefore
x=-8/-8=1
y=--40/-8=5

the answer is
x=1
y=5