Need help if possible

Question

 PRODUCTION SCHEDULING A company has plants located in
Michigan, New York, and Ohio where it manufactures laptop com-
puters, desktop computers, and servers. The number of units of
each product that can be produced per day at each plant are given
in the table below. The company has orders for 2,150 laptop com-
puters, 2,300 desktop computers, and 2,500 servers. How many
days should the company operate each plant in order to exactly ?ll
these orders?

Plant      Mich   NY   Ohio  

Laptop     10    70    60

Desktop  20     50   80

Server   40     30   90

Answer 1

M=Michigan, NY=New York, O=Ohio

10L+20D+40S produced by Michigan per day where L=laptop, D=desktop, S=server

70L+50D+30S produced by New York per day

60L+80D+90S produced by Ohio per day

x=# of days M operates

y=# of days NY operates

z=# of days O operates

10Lx+20Dx+40Sx=total devices produced by M per day

70Ly+50Dy+30Sy=total devices produced by NY per day

60Lz+80Dz+90Sz=total devices produced by O per day

add vertically for total Ls, total Ds, and total Ss

10x+70y+60z=2150 laptops

20x+50y+80z=2300 desktops

40x+30y+90z=2500 servers

we have 3 equations in 3 unknowns to solve

divide all three equations on both sides by 10 to simplify the equations

x+7y+6z=215 (equation 1)

2x+5y+8z=230 (equation 2)

4x+3y+9z=250 (equation 3)

multiply equation 1 by 2

2x+14y+12z=430 (new equation 1) (now we are going to subtract 2 from the new 1)

-

2x+5y+8z+230 (equation 2)

9y+4z=200

now multiply equation 2 by 2 also

4x+10y+16z=460 (new equation 2) (now we are going to subtract 3 from the new 2)

-

4x+3y+9z=250

7y+7z=210

divide by 7 to simplify

y+z=30 and therefore z=30-y

substitute 30-y for z in the equation 9y+4z=200

9y+4(30-y)=200

9y+120-4y=200

5y=200-120

5y=80

y=80/5

y=16 days that NY operates

remember z=30-y ?

z=30-16=14 days that O operates

using equation 1...

x+7y+6z=215 where we found that y=16 and z=14

x+112+84=215

x+196=215

x=215-196

x=19 days that M operates

Michigan operates for 19 days

New York operates for 16 days

Ohio operates for 14 days

we'll do one check and you can do the other two if you like..

19*10+16*70+14*60=190+1120+840=2150 laptops