systems of three variable equations

[cward22]

here is my question:
a certain company charges $3 for shipping orders up to 10kg. it charges $5 for packages from 10kg up to 15kg, and $8 for packages of 15kg or more. one day shipping charges for 200 orders totaled $1105, but the individual receipts were lost. can you figure out the number of packages at each rate if it is known that the number of small packages (less than 10kg) was four-fifths the number of large packages (greater than or equal to 15kg)?

equations i have thus far are:

x + y + z = 200
3x + 5y + 8z = 1105
x=.8z ?????

i figured the answers the hard way by hand and many calculations but i need to know if i am on the right track here and is equation x=.8z correct so i may do this and show my work.
thank you so much!

 

[BigGlenntheHeavy]

\(\displaystyle Use \ rref \ (reduced \ row \ echelon \ form).\)

\(\displaystyle x, \ y, \ z\)

\(\displaystyle 1, \ 1, \ 1, \ = \ 200\)

\(\displaystyle 3, \ 5, \ 8, \ = \ 1105\)

\(\displaystyle 1, \ 0, \ -.8, \ = \ 0\)

\(\displaystyle Hence, \ x \ = \ 60, \ y \ = \ 65, \ and \ z \ = \ 75,\)

\(\displaystyle Check: \ 60+65+75 \ = \ 200, \ 3(60)+5(65)+8(75) \ = \ 1105, \ and \ \frac{60}{75} \ = \ \frac{4}{5}.\)

 

[Denis]

x + y + z = 200
3x + 5y + 8z = 1105
x=.8z ?????

Those equations are correct. Substitute:
.8z + y + z = 200
y + 1.8z = 200 [1]

2.4z + 5y + 8z = 1105
5y + 10.4z = 1105 [2]

Use [1] and [2] to get y and z.

 

[cward22]

why is the .8 negative in the matrix???
thank you both!

 

[masters]

why is the .8 negative in the matrix???
thank you both!

That matrix row comes from \(\displaystyle x=.8z\)

which is the same as \(\displaystyle x+0y-.8z=0\)

Thus, the row: \(\displaystyle 1 \:\: \: 0 \:\:\:-.8\)