Area of a Region You own a triangular tract of land, as shown in the figure. To estimate

Fill in the blanks. The method of using determinants to solve a system of linear equations is called ________ ________.

Fill in the blanks. Three points are ________ if the points lie on the same line.

Fill in the blanks. The area of a triangle with vertices and is given by ________.

Fill in the blanks. A message written according to a secret code is called a ________.

Fill in the blanks. To encode a message, choose an invertible matrix and multiply the ________ row matrices by (on the right) to obtain ________ row matrices.

In Exercises 110, use Cramers Rule to solve (if possible) the system of equations.

In Exercises 110, use Cramers Rule to solve (if possible) the system of equations.

In Exercises 110, use Cramers Rule to solve (if possible) the system of equations.

In Exercises 110, use Cramers Rule to solve (if possible) the system of equations.

In Exercises 110, use Cramers Rule to solve (if possible) the system of equations.

In Exercises 1114, use a graphing utility and Cramers Rule to solve (if possible) the system of equations.

In Exercises 1114, use a graphing utility and Cramers Rule to solve (if possible) the system of equations.

In Exercises 1114, use a graphing utility and Cramers Rule to solve (if possible) the system of equations.

In Exercises 1114, use a graphing utility and Cramers Rule to solve (if possible) the system of equations.

In Exercises 1524, use a determinant and the given vertices of a triangle to find the area of the triangle.

In Exercises 1524, use a determinant and the given vertices of a triangle to find the area of the triangle.

In Exercises 1524, use a determinant and the given vertices of a triangle to find the area of the triangle.

In Exercises 1524, use a determinant and the given vertices of a triangle to find the area of the triangle.

In Exercises 1524, use a determinant and the given vertices of a triangle to find the area of the triangle.

In Exercises 1524, use a determinant and the given vertices of a triangle to find the area of the triangle.

In Exercises 1524, use a determinant and the given vertices of a triangle to find the area of the triangle.

In Exercises 1524, use a determinant and the given vertices of a triangle to find the area of the triangle.

In Exercises 1524, use a determinant and the given vertices of a triangle to find the area of the triangle.

In Exercises 1524, use a determinant and the given vertices of a triangle to find the area of the triangle.

In Exercises 25 and 26, find a value of such that the triangle with the given vertices has an area of 4 square units.

In Exercises 25 and 26, find a value of such that the triangle with the given vertices has an area of 4 square units.

In Exercises 27 and 28, find a value of such that the triangle with the given vertices has an area of 6 square units.

In Exercises 27 and 28, find a value of such that the triangle with the given vertices has an area of 6 square units.

Area of a Region A large region of forest has been infested with gypsy moths. The region is roughly triangular, as shown in the figure. From the northernmost vertex of the region, the distances to the other vertices are 25 miles south and 10 miles east (for vertex ), and 20 miles south and 28 miles east (for vertex ). Use a graphing utility to approximate the number of square miles in this region.

Area of a Region You own a triangular tract of land, as shown in the figure. To estimate the number of square feet in the tract, you start at one vertex, walk 65 feet east and 50 feet north to the second vertex, and then walk 85 feet west and 30 feet north to the third vertex. Use a graphing utility to determine how many square feet there are in the tract of land.

In Exercises 3136, use a determinant to determine whether the points are collinear.

In Exercises 3136, use a determinant to determine whether the points are collinear.

In Exercises 3136, use a determinant to determine whether the points are collinear.

In Exercises 3136, use a determinant to determine whether the points are collinear.

In Exercises 3136, use a determinant to determine whether the points are collinear.

In Exercises 3136, use a determinant to determine whether the points are collinear.

In Exercises 37 and 38, find such that the points are collinear.

In Exercises 37 and 38, find such that the points are collinear.

In Exercises 3944, use a determinant to find an equation of the line passing through the points.

In Exercises 3944, use a determinant to find an equation of the line passing through the points.

In Exercises 3944, use a determinant to find an equation of the line passing through the points.

In Exercises 3944, use a determinant to find an equation of the line passing through the points.

In Exercises 3944, use a determinant to find an equation of the line passing through the points.

In Exercises 3944, use a determinant to find an equation of the line passing through the points.

In Exercises 45 and 46, find the uncoded 1 3 row matrices for the message. Then encode the message using the encoding matrix. TROUBLE IN RIVER CITY

In Exercises 45 and 46, find the uncoded 1 3 row matrices for the message. Then encode the message using the encoding matrix. PLEASE SEND MONEY A

In Exercises 4750, write a cryptogram for the message using the matrix A CALL AT NOON

In Exercises 4750, write a cryptogram for the message using the matrix A ICEBERG DEAD AHEAD

In Exercises 4750, write a cryptogram for the message using the matrix A HAPPY BIRTHDAY

In Exercises 4750, write a cryptogram for the message using the matrix A OPERATION OVERLOAD

In Exercises 5154, use to decode the cryptogram 11 21 64 112 25 50 29 53 23 46 40 75 55 92

In Exercises 5154, use to decode the cryptogram 136 58 173 72 120 51 95 38 178 73 70 28 242 101 115 47 90 36 115 49 199 82

In Exercises 5154, use to decode the cryptogram 9 38 28 25 41 21 31

In Exercises 5154, use to decode the cryptogram 112 140 83 19 25 13 72 76 61 95 _118 71 20 21 38 35 _23 36 42 _48 32

In Exercises 55 and 56,decode the cryptogram by using the inverse of the matrix A 20 17 1 62 143 181

In Exercises 55 and 56,decode the cryptogram by using the inverse of the matrix A 13 61 112 106 11 24 29 65 144 172

The following cryptogram was encoded with a matrix. 8 21 5 10 5 25 5 19 6 20 40 1 16 The last word of the message is _RON. What is the message?

Data Analysis: Supreme Court The table shows the numbers of U.S. Supreme Court cases waiting to be tried for the years 2000 through 2002. (Source: Office of the Clerk, Supreme Court of the United States) (a) Use the technique demonstrated in Exercises 6770 in Section 7.3 to create a system of linear equations for the data. Let represent the year, with corresponding to 2000. (b) Use Cramers Rule to solve the system from part (a) and find the least squares regression parabola (c) Use a graphing utility to graph the parabola from part (b). (d) Use the graph from part (c) to estimate when the number of U.S. Supreme Court cases waiting to be tried will reach 10,000. True or False?

True or False? In Exercises 5961,determine whether the statement is true or false. Justify your answer. In Cramers Rule, the numerator is the determinant of the coefficient matrix.

True or False? In Exercises 5961,determine whether the statement is true or false. Justify your answer. You cannot use Cramers Rule when solving a system of linear equations if the determinant of the coefficient matrix is zero.

True or False? In Exercises 5961,determine whether the statement is true or false. Justify your answer. In a system of linear equations, if the determinant of the coefficient matrix is zero, the system has no solution.

Writing At this point in the text, you have learned several methods for solving systems of linear equations. Briefly describe which method(s) you find easiest to use and which method(s) you find most difficult to use.

In Exercises 6366, use any method to solve the system of equations.

In Exercises 6366, use any method to solve the system of equations.

In Exercises 6366, use any method to solve the system of equations.

In Exercises 6366, use any method to solve the system of equations.

In Exercises 67 and 68, sketch the region determined by the constraints. Then find the minimum and maximum values of the objective function and where they occur, subject to the constraints. Objective function:

In Exercises 67 and 68, sketch the region determined by the constraints. Then find the minimum and maximum values of the objective function and where they occur, subject to the constraints. Objective function: Constraints: Constraints: 6x _ y 40 x _ 3y 15