# A manufacturing company makes two types of water skis, a trick ski and a slalom ski.  The trick ski requires 7 labor-hours for fabricating and 1 labor-hour for finishing.  The slalom ski requires...

A manufacturing company makes two types of water skis, a trick ski and a slalom ski.  The trick ski requires 7 labor-hours for fabricating and 1 labor-hour for finishing.  The slalom ski requires 3 labor-hours for fabricating and 1 labor-hour for finishing.  The maximum labor-hours available per day for fabricating and finishing are 84 and 22, respectively.  If x is the number of trick skis and y is the number of slalom skis produced per day, write a system of linear inequalities that indicates appropriate restraints on x and y.  Find the set of feasible solutions graphically for the number of each type of ski that can be produced.

samhouston | Certified Educator

Trick skis:  x
7 hours fabricating
1 hour finishing

Slalom skis:  y
3 hours fabricating
1 hour finishing

Fabricating max = 84
trick fab hours + slalom fab hours = 84
7 * trick + 3 * slalom = 84
7x + 3y = 84

Finishing max = 22
trick fin hours + slalom fin hours = 22
1 * trick + 1 * slalom = 22
1x + 1y = 22

System:
7x + 3y = 84
x + y = 22

There are several ways to solve the system.  One way is to use substitution.  Solve the second equation for x.

x + y = 22
x + y + (-y) = 22 + (-y)
x = 22 - y

Next, substitute (22 - y) in for x in the first equation.  Then solve for y.

7x + 3y = 84
7(22 - y) + 3y = 84
154 - 7y + 3y = 84
154 - 4y = 84
154 - 4y + (-154) = 84 + (-154)
-4y = -70
-4y / -4 = -70 / -4
y = 17.5

Now substitute 17.5 in for y.

x = 22 - y
x = 22 - 17.5
x = 4.5

The company can make a maximum of 4 trick skis and 17 slalom skis.

Graph below:

If you consider the system as inequalities because the maximum number is given, then any combination of whole-number coordinates will satisfy the system within the quadrilateral formed by the following points:
(0, 22)     (0, 0)     (12, 0)     (4.5, 17.5)