Construct the augmented matrix that corresponds to the following system of equations.5+(8x)/5=y2z−3(x−3y)=02x−y=3(x−4z)

Step 1:To determine:
Given:
System of equations:
${\displaystyle 5+\frac{8x}{5}=y}$
$2z-3(x-3y)=0$
$2x-y=3(x-4z)$
To determine:
The augmented matrix for the given system of equations.
Step 2:Calculation
Consider the equation ${\displaystyle 5+\frac{8x}{5}=y}$
${\displaystyle \Rightarrow \frac{25+8x}{5}=y}$
${\displaystyle \Rightarrow 25+8x=5y}$
${\displaystyle \Rightarrow 8x-5y+0z=-25}$...(1)
Consider the equation $2z-3(x-3y)=0$
${\displaystyle \Rightarrow 2z-3x+9y=0}$
${\displaystyle \Rightarrow -3x+9y+2z=0}$...(2)
Consider the equation $2x-y=3(x-4z)$
${\displaystyle \Rightarrow 2x-y=3x-12z}$
${\displaystyle \Rightarrow 2x-3x-y+12z=0}$
${\displaystyle \Rightarrow -x-y+12z=0}$...(3)
From equation 1,2,3, the augmented matrix is given by:
${\displaystyle \left[\begin{array}{cccc}9& -5& 0& -25\\ -3& 9& 2& 0\\ -1& -1& 12& 0\end{array}\right]}$