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