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.

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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)

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