Creating a system of linear equations

Question

A bookkeeper is trying to determine the prices that a manufacturer was charging by examining old sales slips, which show the number of various items shipped and the total price. He finds that $20$ armchairs, $16$ sofa beds, and $12$ double beds cost $19,600$; that $15$ armchairs, $12$ sofa beds, and $9$ double beds cost $15,700$; and that $12$ armchairs, $10$ sofa beds, and $6$ double beds cost $11,600$.

Could anyone help me start off with creating the system of linear equations for this question? I am little confused as to how to start that, thanks!

Answer 1

we can set the costs of armchair by $x$ the sofa bed by $y$ and the double beds by $z$ so we get the system
$20x+16y+12z=19600$
$15x+12y+9z=15700$
$12x+10y+6z=11600$

Answer 2

HINT: Set variables for the different items sold then write down the the equations for the sales

armchairs $a$

Sofa beds $s$

Double beds $d$

thus the first equation is $20a + 16s+ 12d = 19600$

you should now be able to find the other 2 equations

Answer 3

put in your matrix in the first row the infrmation you get from the first statement (20 armchairs, 16 sofa beds, and 12 double beds cost 19,600),

in the second row the information out of your second statement (15 armchairs, 12 sofa beds, and 9 double beds cost 15,700)

and the same thing with the third row. Now you should get:

$ \begin{pmatrix} 20 & 16 & 12 \\ 15 & 12 & 9 \\ 12 & 10 & 6 \\ \end{pmatrix} $ * $ \begin{pmatrix} x \\ y \\ z \\ \end{pmatrix} $ = $ \begin{pmatrix} 19'600 \\ 15'700 \\ 11'600\\ \end{pmatrix} $

with x: the price of the armchair, y the price of the sofa bed, and z the price of the double bed