2.1 Solution of Linear Systems by the Echelon Method
Suppose that an animal feed is made from two ingredients: corn and soybeans. One serving of each ingredient provides the number of grams of protein and fiber sown in the table. Now suppose we want to know how many servings of corn and soybeans should be used to make a feed that contains 115g of protein and 95g of fiber. Let x represent the number of servings of corn used and y the number of servings of soybeans. First degree equation in n unknowns
Equivalent system Echelon method Back substitution
Ex 1) Solve the system of equations 2x + 3y = 12 3x − 4y = 1
Ex 2) Solve the system of equations 2x - 3y = 6 -4x + 6y= 8
Parameter Ex 3) Solve the system of equations 3x - y = 4 -6x + 2y= -8
Ex 4) Solve the system of equations 2x + y – z = 2 x + 3y + 2z= 1 x + y + z= 2
Ex 5) A flight leaves New York at 8 P.M. and arrives in Paris at 12 Noon (Paris time). This 16-hour difference includes the flight time plus the change in time zones. The return flight leaves Paris at 1 P.M. and arrives in New York at 3 P.M. (NY time). This 2-hour difference includes the flight time minus time zones and no wind. Find the actual flight time eastward and the difference in time zones.
Ex 6) A restaurant owner orders a replacement set of knives, forks, and spoons. The box arrives containing 40 utensils and weighing 141.3 oz (ignoring the weight of the box). A knife, fork, and spoon weigh 3.9 oz, 3.6 oz, and 3.0 oz, respectively. a. How many solutions are there for the number of knives, forks, and spoons in the box? b. Find the solution with the smallest number of spoons.
2.2 Solution of Linear Systems by the Gauss-Jordan Method
2x + y – z = 2 x + 3y + 2z= 1 x + y + z= 2 Augmented matrix Gauss-Jordan method
Ex 1) Use the Gauss-Jordan method to solve the system 4x + 5y = 10 7x + 8y = 19
Ex 2) Use the Gauss-Jordan method to solve the system x + 5z = -6 + y 3x +3y = 10 + z x + 3y + 2z = 5
Ex 3) Use the Gauss-Jordan method to solve the system x – 2y = 2 3x – 6y = 5
Ex 4) Use the Gauss-Jordan method to solve the system x + 2y – z = 0 3x – y + z = 6 -2x – 4y + 2z = 0
Ex 5) Use the Gauss-Jordan method to solve the system x + 2y + 3z – w = 4 2x + 3y + w = -3 3x + 5y + 3z = 1
Ex 6) A convenience store sells 23 sodas one summer afternoon in 12, 16, and 20 oz cups (small, medium, and large). The total volume of soda sold was 376 oz. • Suppose that the prices for a small, medium, and large soda are $1, $1.25, and $1.40, respectively, and that the total sales were $28.45. How many of each size did the store sell?
b. Suppose the prices for small, medium, and large sodas are changed to $1, $2, and $3, respectively, but all other information is kept the same. How many of each size did the store sell? c. Suppose the prices are the same as in part (b), but the total revenue is $48. Now how many of each size did the store sell?
2.3 Addition and Subtraction of Matrices
Ex 1) The EZ Life Company manufactures sofas and armchairs in three models, A, B, and C. The company has regional warehouses in New York, Chicago, and San Francisco. In its August shipment, the company sends 10 model-A sofas, 12 model-B sofas, 5 model-C sofas, 15 model-A chairs, 20 model-B chairs, and 8 model-C chairs to each warehouse. Use a matrix to organize this information.
Ex 2) Give the size of each matrix. Square matrix Row matrix / row vector Column matrix / column vector Ex 3) State the conditions present for each statement to be true.
Ex 5) The September shipments from the EZ Life Company to the New York, San Francisco, and Chicago warehouses are given in matrices N, S, and C below. What was the total amount shipped to the three warehouses in September?
2.4 Multiplication of Matrices
Ex 2) Find the product AB of matrices Ex 3) Find the product CD of matrices
Ex 4) Suppose matrix A is 2x2 and matrix B is 2x4. Can the products AB and BA be calculated? If so, what is the size of each product? Ex 5) Find AB and BA, given
Ex 6) A contractor builds three kinds of houses, models A, B, and C, with a choice of two styles, Spanish and contemporary. Matrix P shows the number of each kind of house planned for a new 100-home subdivision. The amounts for each of the exterior materials depend primarily on the style of the house. These amounts are shown in matrix Q. ( Concrete is in cubic yards, lumber in units of 1000 board feet, brick in 1000s, and shingles in units of 100 ft2.) Matrix R gives the cost in dollars for each kind of material. a. What is the total cost of these materials for each model?
b. How much of each of the four kinds of material must be ordered? c. What is the total cost for exterior materials? d. Suppose the contractor builds the same number of homes in five subdivisions. Calculate the total amount of each exterior material for each model for all five subdivisions.
2.5 Matrix Inverses
Identity matrix Multiplicative inverse matrix Coefficient matrix Ex 1) Verify that the matrices are inverses of each other.
Ex 4) Use the inverse of the coefficient matrix to solve the linear system 2x – 3y = 4 x + 5y = 2
Ex 5) Three brands of fertilizer are available that provide nitrogen, phosphoric acid and soluble potash to the soil. One bag of each brand provides the units of each nutrient shown in the table. For ideal growth, the soil on a Michigan farm needs 18 units of nitrogen, 23 units of phosphoric acid, and 13 units of potash per acre. The corresponding numbers for a California farm are 31, 24, and 11, and for a Kansas farm are 20, 19, and 15. How many bags of each brand of fertilizer should be used per acre for ideal growth on each farm?