Answer to Question #169916 in Math for Mythaiuz Mohan

Question

A leading process engineering company bought three (3) major pieces of equipment:

a shell and tube heat exchanger; a storage tank and a three-phase vertical separator.

The company purchased the exact same type of equipment three (3) times within a

ten (10) year period.

Due to audit concerns and missing receipts, you are tasked with determining the cost of EACH

piece of equipment in $US, using The Gauss-Jordan Matrix Method. You are only provided with the total cost(US$) of the following:

1st Time: 3 tanks + 4 heat exchangers + 2 vertical separators cost US$157,500

2nd Time: 2 tanks + 1 heat exchanger + 3 vertical separators cost US$90,000

3rd Time: 5 tanks + 3 heat exchangers + 2 vertical separators cost US$149,500

Answer 1

Let a tank costs $x$ $US, a heat exchanger costs $y$ $US, and a vertical separator costs $z$ $US. Then

$$\begin{matrix} 3x+4y+2z=157500 \\ 2x+y+3z=90000 \\ 5x+3y+2z=149500 \\ \end{matrix}$$

Then

$$A=\begin{pmatrix} 3 & 4 & 2 & & 157500 \\ 2 & 1 & 3 & & 90000 \\ 5 & 3 & 2 & & 149500 \\ \end{pmatrix}$$

Divide row 1 by 3 $(R_1=\dfrac{R_1}{3})$

$$\begin{pmatrix} 1 & 4/3 & 2/3 & & 52500 \\ 2 & 1 & 3 & & 90000 \\ 5 & 3 & 2 & & 149500 \\ \end{pmatrix}$$

Subtract row 1 multiplied by 2 from row 2 $(R_2=R_2-(2)R_1)$

$$\begin{pmatrix} 1 & 4/3 & 2/3 & & 52500 \\ 0& -5/3 & 5/3 & & -15000 \\ 5 & 3 & 2 & & 149500 \\ \end{pmatrix}$$

Subtract row 1 multiplied by 5 from row 3 $(R_3=R_3-(5)R_1)$

$$\begin{pmatrix} 1 & 4/3 & 2/3 & & 52500 \\ 0& -5/3 & 5/3 & & -15000 \\ 0 & -11/3& -4/3 & & -113000 \\ \end{pmatrix}$$

Multiply row 2 by $-\dfrac{3}{5}$ $(R_2=(-\dfrac{3}{5})R_2)$

$$\begin{pmatrix} 1 & 4/3 & 2/3 & & 52500 \\ 0 & 1 &-1 & & 9000 \\ 0 & -11/3& -4/3 & & -113000 \\ \end{pmatrix}$$

Subtract row 2 multiplied by $\dfrac{4}{3}$ from row 1 $(R_1=R_1-(\dfrac{4}{3})R_2)$

$$\begin{pmatrix} 1 & 0 & 2 & &40500 \\ 0 & 1 &-1 & & 9000 \\ 0 & -11/3& -4/3 & & -113000 \\ \end{pmatrix}$$

Add row 2 multiplied by $\dfrac{11}{3}$ to row 3 $(R_3=R_3+(\dfrac{11}{3})R_2)$

$$\begin{pmatrix} 1 & 0 & 2 & &40500 \\ 0& 1 &-1 & & 9000 \\ 0 & 0 & -5 & & -80000 \\ \end{pmatrix}$$

Divide row 3 by $-5$ $(R_3=\dfrac{R_3}{-5})$

$$\begin{pmatrix} 1 & 0 & 2 & &40500 \\ 0& 1 &-1 & & 9000 \\ 0 & 0 & 1 & & 16000 \\ \end{pmatrix}$$

Subtract row 3 multiplied by 2 from row 1 $(R_1=R_1-(2)R_3)$

$$\begin{pmatrix} 1 & 0 & 0 & &8500 \\ 0& 1 &-1 & & 9000 \\ 0 & 0 & 1 & & 16000 \\ \end{pmatrix}$$

Add row 3 to row 2 $(R_2=R_2+R_3)$

$$\begin{pmatrix} 1 & 0 & 0 & &8500 \\ 0& 1 & 0 & & 25000 \\ 0 & 0 & 1 & & 16000 \\ \end{pmatrix}$$

A tank costs $8500$ $US, a heat exchanger costs $25000$ $US, and a vertical separator costs $16000$ $US.