Answer to Question #334688 in Linear Algebra for fade

Question

c) Martin, Arop and Sam and went shopping at Kabale mega super market buying similar items. Martin bought one kilogram of sugar and one tin of blue band spending 6 shillings altogether. Arop bought bought 3 kilograms of maize flour and one tin of blue band spending 17 shillings altogether. Sam bought two kilograms of sugar, one kilogram of maize flour and 3 tins of blue band spending 15 shillings altogether. i) Express their purchases as equations ii) Using matrix method, find the cost of each kilogram of sugar, maize flour and a tin of blue band.

Answer 1

x = one kilogram of sugar

y = one tin of blue band

z = one kilogram of maize

$\begin{cases} x+y=6\\ y+3z = 17\\ 2x +3y+z = 15 \end{cases}$

We rewrite the system of equations in matrix form and solve it using the Gauss method

$\begin{pmatrix} 1 & 1 & 0 \:\: |6\:\:\\ 0 & 1 & 3 \:\:| 17 \\ 2 & 3 & 1\:\:| 6\:\: \end{pmatrix}\\$

from line 3 subtract 1 line multiplied by 2

$\begin{pmatrix} 1 & 1 & 0 \:\: |6\:\:\\ 0 & 1 & 3 \:\:| 17 \\ 0 & 1 & 1\:\:| 3\:\: \end{pmatrix}\\$

from line 1 we subtract line 2, multiplied by 1; from line 3 subtract line 2 multiplied by 1

$\begin{pmatrix} 1 & 0 & -3 \: |-11\\ 0 & 1 & 3 \:|\:\: 17 \\ 0 & 0 & -2\:| -14 \end{pmatrix}\\$

Divide the 3rd row by -2

$\begin{pmatrix} 1 & 0 & -3 \: |-11\\ 0 & 1 & 3 \:|\:\: 17 \\ 0 & 0 & 1\:|\:\:\:\: 7 \end{pmatrix}\\$

to line 1 add line 3 multiplied by 3; from line 2 subtract line 3 multiplied by 3

$\begin{pmatrix} 1 & 0 & 0 \: |\:\:\:10\\ 0 & 1 & 0 \:| -4 \\ 0 & 0 & 1\:|\:\:\:\:\: 7 \end{pmatrix}\\$

twet

$\begin{cases} 10\\ -4\\ 7 \end{cases}$

ANSWER:

x = 10

y = -4

z = 7

BUT "y" cannot be negative