Objective 4.1A
Vocabulary to Review
equation in two variables [3.1A]
New Vocabulary
system of equations
solution of a system of equations in two variables
independent system of equations
inconsistent system of equations
dependent system of equations
Discuss the Concepts
For a system of two linear equations in two variables, explain, in geometric terms, each of the following: dependent system of equations, independent system of equations, and inconsistent system of equations.
Concept Check
1. How is the solution of a system of equations in two variables represented?
2. Can a system of two linear equations in two variables have exactly two solutions? Explain your answer.
Optional Student Activity
For each system given below, graph the equations and label the point of intersection. Then add the two equations in the system and graph the resulting equation on the same coordinate system on which you graphed the system of equations. How is the graph of the sum of the equations related to the graph of the system of equations?
1. 3x – 4y = 12
5x + 4y = -12
2. 2x – 3y = 6
-2x + 5y = -10
3. 2x – 3y = 2
5x + 4y = 5
4. 2x – y = 1
x + 2y = 3
The graph of the sum of the equations is a line whose graph contains the point of intersection of the graph of the system of equations.
Objective 4.1B
New Vocabulary
substitution method
Discuss the Concepts
To introduce the substitution method, you might ask students to solve the following systems of equations.
1. 3x – 2y = 4
x = 2 (2, 1)
2. y = -2
2x + 3y = 4 (5, -2)
Ask the students to explain why we can substitute 2 for x in the first example and -2 for y in the second example. Then ask them to use substitution to solve the following system.
3. y = 2x + 1
x + 2y = 3 (1, 1)
Ask them to explain why we can substitute 2x + 1 for y in this situation.
Concept Check
When you solve a system of equations by the substitution method, how do you determine whether the system of equations is inconsistent? How do you determine whether it is dependent?
Optional Student Activity
1. Write a system of equations for each case.
a. The system has (-3, 5) as its only solution.
Answers will vary. For example,
y = x + 8
y = -x + 2
b. The system has no solution.
Answers will vary. For example,
y = 2x – 3
y = 2x + 4
c. The system is a dependent system of equations.
Answers will vary. For example,
y = 3x – 5
3x - y = 5
2. For what value of k is the system of equations inconsistent?
a.
2
b.
1
c.
2
3. Explain how to determine whether a linear system of equations in two variables is independent, inconsistent, or dependent without graphing or solving the system of equations.
Answers will vary. For example, write each equation in the system in the form y = mx + b. If the equations have the same slope and the same y-intercept, the system of equations is dependent. If the equations have the same slope and different y-intercepts, the system of equations is inconsistent. If the equations in the system do not have the same slope, the system is independent.
Objective 4.1C
New Formulas
Principal ∙ interest rate = interest earned (Pr = I)
Discuss the Concepts
1. Explain what each variable in the formula Pr = I represents.
2. What is the difference between interest and interest rate?
3. For the following example, name (a) the principal, (b) the interest rate, and (c) the interest earned. The annual simple interest rate on a $1250 investment is 5%. Find the annual simple interest earned on the investment.
Optional Student Activity
1. A bank offers customers a 2-year certificate of deposit (CD) that earns 8% compound annual interest. This means that the interest earned each year is added to the principal before the interest for the next year is calculated. Find the value after 2 years of a nurse’s investment of $2500 in this CD. $2916
2. A bank offers customers a 3-year certificate of deposit (CD) that earns 8.5% compound annual interest. This means that the interest earned each year is added to the principal before the interest for the next year is calculated. Find the value after 3 years of an accountant’s investment of $3000 in this CD. $3831.87
Objective 4.2A
Vocabulary to Review
system of equations in two variables [4.1A]
New Vocabulary
addition method
Discuss the Concepts
Ask students to explain how the following situation is related to an inconsistent system of equations.
The perimeter of a rectangle is 100 m. The sum of the length and width of the rectangle is 40 m. Find the dimensions of the rectangle.
Concept Check
Mark said to his brother John, “Give me eight of your transformers and then we’ll each have the same number.” John answered, “No, you give me eight of your transformers and then I will have twice as many as you.” How many transformers did John have to start with? 56 transformers
Optional Student Activity
1. The point of intersection of the graphs of the equations Ax + 2y = 2 and 2x + By = 10 is (2, -2). Find A and B. A = 3, B = -3
2. The point of intersection of the graphs of the equations Ax – 4y = 9 and 4x + By = -1 is (-1, -3). Find A and B. A = 3, B = -1
3. Given that the graphs of the equations 2x – y = 6, 3x – 4y = 4, and Ax – 2y =0 all intersect at the same point, find A. A = 1
4. Given that the graphs of the equations 3x – 2y = -2, 2x – y = 0 , and Ax + y = 8 all intersect at the same point, find A. A = 2
5. Find an equation such that the system of equations formed by your equation and 2x – 5y = 9 will have (2, -1) as a solution. Answers will vary. 3x + 2y = 4 is a possibility.
Objective 4.2B
Vocabulary to Review
plane [3.1A]
independent system of equations [4.1A]
inconsistent system of equations [4.1A]
dependent system of equations [4.1A]
New Vocabulary
linear equation in three variables
xyz-coordinate system
ordered triple
solution of an equation in three variables
system of linear equations in three variables
solution of a system of equations in three variables
Discuss the Concepts
A good model of a three-dimensional coordinate system is the corner of the floor in a room. The xy-plane is the floor, the xz-plane is one wall, and the yz-plane is the other wall.
Give students various ordered triples with positive coordinates and have them indicate the location of the point in the room. Then ask where a point that has negative numbers as some or all of its coordinates would be located. Finally, ask students to identify the region for which (1) x = 0 (2) y = 0 (3) z = 0, (4) x and y are both 0, (5) x and z are both 0, (6) y and z are both 0, and (7) x, y, and z are 0.
Concept Check
1. The sum of three numbers is 62. The sum of the first and second numbers is equal to 10 less than the third. The first number minus the second number is equal to 34 less than the third number. Find the three numbers. 14, 12, 36
2. Jason and his older brother Aaron together have $62. Jason and his sister Michelle together have $67. Michelle and Aaron together have $29. What is the least amount of money any of the three siblings has? $12
Optional Student Activity
1. Find a three-digit number such that the sum of the digits is 7, the number is increased by 99 if the digits are reversed, and the hundreds digit is 3 less than the sum of the other two digits. 223
2. Let L be the line along which the planes 2x + y – z = 13 and x – 2y + z = 5 intersect. If the point (x, 3, z) lies on L, find the value of (x – z). 3
Answers to Writing Exercises
67. a. The system of equations has no solution; it is inconsistent. See Figures A, B, C, and D on page 217.
b. The system of equations has exactly one solution; it is an independent system whose solution is a point in space. See Figure E on page 217.
c. The system of equations has infinitely many solutions; it is a dependent system. See Figures F, G, and H on page 217.
68. a. The graph of x = 3 in an xyz-coordinate system is a plane parallel to the yz-plane at x = 3.
b. The graph of y = 4 in an xyz-coordinate system is a plane parallel to the xz-plane at y = 4.
c. The graph of z = 2 in an xyz-coordinate system is a plane parallel to the xy-plane at z = 2.
d. The graph of y = x in an xyz-coordinate system is a vertical plane perpendicular to the xy-plane and 458 from the xz and yz planes.
Objective 4.3A
New Vocabulary
matrix
element of a matrix
order m ´ n
square matrix
determinant
minor of an element
cofactor of an element of a matrix
expanding by cofactors
Discuss the Concepts
1. What is a matrix?
2. What is a square matrix?
3. What is the difference between the notation for a matrix and the notation for a determinant?
4. How do you find the minor of an element in a 3 ´ 3 determinant?
Concept Check
1. Find the value of the expression.
-190
2. If all the elements in one row or one column of a 2 ´ 2 matrix are zeros, what is the value of the determinant of the matrix? 0
3. If all the elements in one row or one column of a 3 ´ 3 matrix are zeros, what is the value of the determinant of the matrix? 0
Optional Student Activity
1. Solve for x.
a. . -1
b. -14
2. a. Find the value of the determinant . 0
b. If two columns of a 3 ´ 3 matrix contain identical elements, what is the value of the determinant? 0
3. The area of a triangle with vertices (x1, y1), (x2, y2), and (x3, y3) can be given as one-half the absolute value of the determinant shown below. Find the area of the triangle whose vertices are (-6, 3), (1, 5), and (4, -5).
38 square units
Objective 4.3B
New Vocabulary
coefficient determinant
numerator determinant
New Formulas
Cramer’s Rule for a System of Two Equations in Two Variables
Cramer’s Rule for a System of Three Equations in Three Variables
Discuss the Concepts
1. When can Cramer’s Rule not be used to solve a system of equations?
2. If the determinant of the denominator is zero when you are using Cramer’s Rule, the system of equations is either dependent or inconsistent. Explain how you can determine which it is.
Concept Check
Find the ordered pair of numbers that satisfies the system
Optional Student Activity
Find the equation of a plane that contains the given points.
1. (2, 1, 1), (-1, 2, 12), (3, 2, 0)
2. (1, -1, 5), (2, -2, 9), (-3, -1, -1)
Answers to Writing Exercises
1. The determinant associated with the matrix is Its value is
2. The cofactor of a given element in a matrix is (-1)i + j times the minor of that element, where i is the row number of the element and j is the column number of the element.
Objective 4.4A
Formulas to Review
[2.3C]
Discuss the Concepts
In rate-of-wind and rate-of-current problems, how does the wind or current affect the rate of the object? (You want students to understand that when an object is traveling with the wind or current, its speed increases; traveling against the wind or current decreases the speed of the object.)
Concept Check
1. The speed of a plane is 500 mph. There is a headwind of 50 mph. What is the speed of the plane relative to an observer on the ground? 450 mph
2. The rate of a current in a river is x mph, and the rate of a boat in still water is y mph.
a. How can you represent the rate of the boat going down the river? y + x
b. How can you represent the rate of the boat going up the river? y – x
Optional Student Activity
A plane is flying 3500 mi from New York City to London. The speed of the plane in calm air is 375 mph, and there is a 50-mph tailwind. The point of no return is the point at which the flight time required to return to New York City is the same as the flight time required to travel on to London. For this flight, how far from New York is the point of no return? Round to the nearest whole number. 1517 mi