Math Problems: Systems of Equations, Optimization and Linear Programming

Solving Systems of Equations using Substitution and Elimination Methods

1. Solve the following system of equations using the substitution method 
    3x - 2y =  10
                          y = -4x + 1

2. Solve the following system of equations using the elimination method.
                5x +  2y = -10
                2x - 3y = 6

3. There are three types of  boats that carry metal products, wood products, and
plastic products. The amount of each type of product, in kilograms, that each boat can carry is given in the following table.
Boat Type 1     Boat Type 2     Boat Type 3
Metal                5000          0             7500
Wood                4000        500             3000
Plastic               0      4500             2800
Set up, but do not solve the system of linear equations that must be solved to
determine the number of each type of boat required to transport 90,000
Kgs of metal, 78,500 Kgs of wood and 40,000 Kgs of plastic from a port in Windsor to a port in Montreal. Be sure to include a well written definition of the variable
4. Set up a system of linear equations to solve the following problem. Be sure to include a well written definition of the variables.Willy B. Rich invests $27,000.00 in 2 accounts this year, account A and account B . Account A pays and annual interest rate of 4.0% on the amount invested and account B pays 6.5% /year . How much should Willy invest in each account this year in order to earn $1380.00
5. Set up or formulate but do not solve the following LP problem. Be sure to include a well written definition for the variables.
 A farmer must plant two crops, wheat and barley. Wheat will yield a profit of $150/acre and barley will yield a profit of $200/acre. Wheat costs $40/acre to produce and harvest and barley costs $60/acre. The farmer has a maximum of $7400.00 available to produce and harvest these two crops. Each acre of wheat requires 20 man-hours and each acre of barley costs 25 man-hours. The farmer must keep the man-hours less than or equal to 3300. How many acres of each crop should he plant in order to maximize his profit?
6. Draw the feasible region and find the corner points for the following system. Label all corners and clearly indicate the required region .
                -x  + y  2
                 x +2y  10
                3x + y  15
                        x  0
                        y  0


7. Solve the following linear programming problem graphically using the method of corners.
 
    Minimize      f = 2x + 5y
    
    Subject to    30x + 25y   400
            x + 0.5y    10
            2x + 5y  40
            x  0
            y   0