SOLUTION: A television manufacturing company makes three models of television sets. The super model requires 5 hours of wiring time, 6 hours of assembly time, and 2 hours of testing time.

Question 1037820: A television manufacturing company makes three models of television sets. The super model requires 5 hours of wiring time, 6 hours of assembly time, and 2 hours of testing time. The deluxe model requires 3 hours of wiring time, 4 hours of assembly time, and 1 hour of testing time. The regular model requires 2 hours of wiring time, 3 hours of assembly time, and 1 hour of testing time. There are 690 hours available for wiring, 910 hours available for assembly, and 280 hours available for testing each month. How many of each type of television set should be manufactured in order to use all the available hours each month?
Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
set up a table as shown below:

              wire       assembly        test
super           5           6             2
deluxe          3           4             1
regular         2           3             1
total           690         910           280

let x represent number of super television sets.
let y represent number of deluxe television sets.
let z represent number of regular television sets.

your constraint equations are:

5x + 3y + 2z = 690 (equation 1)
6x + 4y + 3z = 910 (equation 2)
2x + y + z = 280 (equation 3)

these are 3 simultaneous equations in 3 unknowns that have to be solved simultaneously.

multiply both sides of the equation 1 by 3 and both sides of equation 2 by 2 to get:

15x + 9y + 6z = 2070 (equation 1 * 3)
12x + 8y + 6z = 1820 (equation 2 * 2)

subtract equation 2 * 2 from equation 1 * 3 to get:

3x + y = 250 (equation 4)

leave equation 2 as is and multiply equation 3 by 3 to get:

6x + 4y + 3z = 910 (equation 2)
6x + 3y + 3z = 840 (equation 3 * 3)

subtract equation 3 * 3 from equation 2 to get:

y = 70 (equation 5)

you are left with equations 4 and 5 as shown below:

3x + y = 250 (equation 4)
y = 70 (equation 5)

replace y with 70 in equation 4 to get:

3x + 70 = 250
subtract 70 from both sides of this equation to get:
3x = 180
divide both sides of this equation by 3 to get:
x = 60

so far you have x = 60 and y = 70

go back to any of the original equations and replace x with 60 and y with 70 and solve for z.

we'll use equation 1.

start with:
5x + 3y + 2z = 690 (equation 1)
replace x with 60 and y with 70 to get:
5*60 + 3*70) + 2z = 690
simplify to get:
300 + 210 + 2z = 690
combine like terms to get:
510 + 2z = 690
subtract 510 from both sides of the equation to get:
2z = 690 - 510
simplify to get:
2z = 180
divide both sides of the equation by 2 to get:
z = 90

now you have:
x = 60
y = 70
z = 90

replace x,y,z with 60,70,90 in all 3 equations and they should all be true.

start with:

5x + 3y + 2z = 690 (equation 1)
6x + 4y + 3z = 910 (equation 2)
2x + y + z = 280 (equation 3)

replace x with 60 and y with 70 and z with 90 to get:

5*60 + 3*70 + 2*90 = 690 (equation 1)
6*60 + 4*70 + 3*90 = 910 (equation 2)
2*60 + 70 + 90 = 280 (equation 3)

simplify to get:

300 + + 210 + 180 = 690 (equation 1)
360 + 280 + 270 = 910 (equation 2)
120 + 70 + 90 = 280 (equation 3)

combine like terms to get:

690 = 690 (equation 1)
910 = 910 (equation 2)
280 = 280 (equation 3)

all three equations are true when x = 60 and y = 70 and z = 90, therefore the solution is correct.

since x represents the number of super televisions and y represents the number of deluxe televisions and z represents the number of regular televisions, ....

the company should make 690 super, 910 deluxe, and 280 regular television so that they can utilize all the available resources.

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