Chapter 6 Review Name___________________________________ SHORT ANSWER. Write the word or phrase that best completes each statement or answers the question. Solve the system of two equations in two variables. 1) 5x + 9y = 43 -2x - 7y = -24 1) Multiply both sides of each equation by a common denominator to eliminate the fractions. Then solve the system. 2) 3x 8 - 3y 5 = 33 80 4x 7 + 4y5 = 3735 2) Obtain an equivalent system by performing the stated elementary operation on the system. 3) Replace the third equation by the sum of itself and -1 times the second equation. x - 2y - 7z = 17 -6x + 4y + 5z = -9 8x + 7y - z = -4 3) Solve the system by back substitution. 4) x + 4y+ 4z = 11 3y + 5z = 17 2z = 8 4) Write an augmented matrix for the system of equations. 5) 2x + 9y + 6z = 32 4x + 6y + 4z = 24 3x + 2y + 8z = 63 5) Write the system of equations associated with the augmented matrix. Do not solve. 6) 1 0 0 6 0 1 0 -7 0 0 1 9 6) Perform the row operations on the matrix and write the resulting matrix. 7) Replace R2 by 12R1 + 12R2 2 0 6 7)
9) x + y - 2z = 8 3x + z = - 6 2x - y + 3z = -14 9) Solve the system of equations. If the system is dependent, express solutions in terms of the parameter z. 10) x + y + z = 2 x - y + 5z = 12 5x + y + z = -6 10) 11) 2x + y - z = 2 x - 3y + 2z = 1 7x - 7y + 4z = 7 11) Use the Gauss-Jordan method to solve the system of equations. 12) x + y + z = -12 x - y + 3z = -8 4x + y + z = -24 12) 13)-3x - y - 9z = -75 3x + 5y - 2z = 44 -8x - 6y + z = -95 13) Solve the problem by writing and solving a suitable system of equations. 14) Alan invests a total of $18,000 in three different ways. He invests one part in a mutual fund which in the first year has a return of 11%. He invests the second part in a government bond at 7% per year. The third part he puts in the bank at 5% per year. He invests twice as much in the mutual fund as in the bank. The first year Alanʹs investments bring a total return of $1500. How much did he invest in each way? 14) 15) A company makes 3 types of cable. Cable A requires 3 black, 3 white, and 2 red wires. Cable B requires 1 black, 2 white, and 1 red wires. Cable C requires 2 black, 1 white, and 2 red wires. They used 100 black, 110 white and 80 red wires. How many of each cable were made? 15) 16) A small business takes out loans from three different banks to buy some new equipment. The total amount of the three loans is $19,000. The first bank offered an interest rate of 16%. The second bank offered a rate of 18% and the amount borrowed from this bank was $5000 less than twice as much as the amount borrowed from the first bank. The third bank offered a rate of 15%. The total annual interest was $ 3050. How much did they borrow from each bank? 16)
Solve the problem.
17) According to an agricultural report, the amounts of nitrogen (lb/acre), phosphate (lb/acre), and labor (lb/acre), needed to grow three different vegetables are given by the following tab
Vegetable A Vegetable B Vegetable C Nitrogen 120 150 170 Phosphate 140 100 80 Labor 4.8 4.4 4.6 If a farmer has 260 acres, 36,200 pounds of nitrogen, 30,000 pounds of phosphate, and 1200 hours of labor, can he use all of his resources completely? If so, how many acres should he allot for each crop? 17) 18) What is the size of the matrix? -5 7 -3 18) Perform the indicated operation where possible. 19) -1 5 0 4 8 -4 - 2 17 4 3 2 19) Perform the indicated operation. 20) Let A = 3 3 2 4 and B = 0 4-1 6 . Find 4A + B. 20) Write a matrix to display the information. 21) A bakery sells three types of cakes. Cake I requires 2 cups of flour, 2 cups of sugar, and 2 eggs. Cake II requires 4 cups of flour, 1 cup of sugar, and 1 egg. Cake III requires 2 cups of flour, 2 cups of sugar, and 3 eggs. Make a 3 x 3 matrix showing the required ingredients for each cake. Assign the cakes to the rows and the ingredients to the columns. 21) Find the order of the matrix product AB and the product BA, whenever the products exist. 22) A is 4 x 1, B is 1 x 4. 22) Given the matrices A and B, find the matrix product AB. 23) A = -1 3 3 2 , B = -2 0-1 1 Find AB. 23) 3 0
Determine whether the two matrices are inverses of each other by computing their product. 26) 5 3 3 2 and 2 -3 -3 5 26) 27) 1 2 0 -2 0 2 3 2 -4 and 1 2 -1 - 12 1 4 1 2 1 4 1 2 - 12 - 12 27) Find the inverse, if it exists, of the given matrix. 28) A = 3 6 -5 6 28) 29) A = 1 -3 -5 0 29) 30) -2 1 3 3 -1 2 -4 2 0 30)
