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ANSWER THE FOLLOWING PROBLEMS. SHOW YOUR COMPLETE WORK.

1) A farmer has 20 hectares for growing barley and swedes. The farmer has to decide how
much of each to grow. The cost per hectare for barley is $30 and for swedes is $20. The
farmer has budgeted $480. Barley requires 1 man-day per hectare and swedes require 2
man-days per hectare. There are 36 man-days available. The profit on barley is $100 per
hectare and on swedes is $120 per hectare. Find the number of hectares of each crop the
farmer should sow to maximize profits.
Use the following to answer the problem above:
(a) Unknowns
x = number of hectares of barley
y = number of hectares of swedes
(b) Constraints
Land x + y ≤ 20
Cost 30x + 20y ≤ 480
Manpower x + 2y ≤ 36
Apparent Constraints
x ≥ 0
y ≥ 0
(c) Profit
P = 100x + 120y

2) Solve the system of linear equations in three variables below:
2x − y + z = 4
x + 3y − z = 11
4x + y − z = 14

Respuesta :

Xaioo

Final Answer-Explanation:

To solve the system of linear equations, we can use the method of substitution or elimination. Let's use the method of substitution.

First, we can solve the first equation for x:

2x = y - z + 4

x = (y - z + 4)/2

Next, we can substitute the expression for x into the second and third equations:

(y - z + 4)/2 + 3y - z = 11

4(y - z + 4)/2 + y - z = 14

Simplify the equations:

y/2 - z/2 + 2 + 3y - z = 11

2y - 2z + 8 + y - z = 14

Combine like terms:

5y/2 - 3z/2 + 2 = 11

3y - 3z + 8 + y - z = 14

Now, we can solve the first equation for y:

5y - 3z + 4 = 22

5y - 3z = 18

5y = 3z + 18

y = (3z + 18)/5

Now, we can substitute the expression for y into the second equation:

3(3z + 18)/5 - 3z + 8 + (3z + 18)/5 - z = 14

Simplify the equation and solve for z:

9z/5 + 54/5 - 3z + 8 + 3z/5 + 18/5 - z = 14

9z + 54 - 15z + 40 + 3z + 18 - 5z = 70

-3z + 112 = 70

-3z = 70 - 112

-3z = -42

z = 14

Now that we have z, we can solve for y using the expression we found earlier:

y = (3z + 18)/5

y = (3*14 + 18)/5

y = (42 + 18)/5

y = 60/5

y = 12

Finally, we can solve for x using the expression we found earlier:

x = (y - z + 4)/2

x = (12 - 14 + 4)/2

x = 2/2

x = 1

So the solution to the system of linear equations is:

x = 1

y = 12

z = 14

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