Cap04

Name:

Chapter 4 Pretest Form A

Date:

Determine whether or not each set of ordered pairs is a solution to the system. 2 x − 7 y = 13 5 y = 3x − 14 1. (3, –1)

1. ____________________________

2. (–2, –4)

2. ____________________________

Determine, without solving the system, whether the system of equations is consistent, inconsistent, or dependent. State whether the system has exactly one solution, no solution, or an infinite number of solutions.

2 x−4 3 2x − 3 y = 6

3. y =

3. ____________________________

4. x − 4 y = −5 y = 9 x − 25

4. ____________________________

5. 5 x − 6 y = 8 3 y = 2.5 x − 4

5. ____________________________

Solve the system of equations graphically. 6.

x + y = −2

6.

y

y = 2x − 5

x

Solve each system of equations by substitution. 7. y = x − 4 3x + 2 y = 7

7. ____________________________

8. y = 7 x − 9 y = 5x + 7

8. ____________________________

9. x − 3 y = 1 2 x + y = −5

9. ____________________________

Solve each system of equations by addition. 10.

x + y = −11

10. ____________________________

x − 2 y = 10

11. 2 x − 5 y = 15 4 x + 5 y = −9

11. ____________________________

131

Chapter 4 Pretest Form A (cont.)

Name:

12. Write the augmented matrix for the following system of equations. 5 x + y = −9 3x − 2 y = 7

12. ____________________________

13. Evaluate the determinant. 14 −5

13. ____________________________

2

−1

Refer the following for problems 14–15. The sum of two numbers is 9. The larger number is 3 more than twice the smaller. 14. Write a system of equations you could use to find the two numbers.

14. ____________________________

15. Solve the system and find the numbers.

15. ____________________________

16.

Determine the solution to the system of inequalities. 4 x − 2 y ≥ −8 3 x + 4 y < 12

16.

y

x

132

Name:

Chapter 4 Pretest Form B

Date:

Determine whether or not each set of ordered pairs is a solution to the system. x + 2y = 4 2 x + y = −1 1. (–2, 3)

1. ____________________________

2. (2, 1)

2. ____________________________

Determine, without solving the system, whether the system of equations is consistent, inconsistent, or dependent. State whether the system has exactly one solution, no solution, or an infinite number of solutions. 3. 2 x − 5 y = −19 5x − 2 y = 5

3. ____________________________

4. 3 x − 4 y = 36 3 y = x−9 4

4. ____________________________

5. 10 x − 4 y = 6 5 y = x −1 2

5. ____________________________

Solve the system of equations graphically. 6.

3 x − 2 y = −12

6.

y

y = −3 x − 3

x

Solve each system of equations by substitution. 7. y = x − 3 2 x − 5 y = −6

7. ____________________________

8. y = 9 x − 3 y = 7x + 5

8. ____________________________

9. x − 4 y = −7 3 x + y = −8

9. ____________________________

Solve each system of equations by addition. 10. 5 x − y = 16 2x + y = 5

10. ____________________________

11. 4 x − 3 y = 10 x + 3 y = 15

11. ____________________________

133

Chapter 4 Pretest Form B (cont.)

Name:

12. Write the augmented matrix for the following system of equations. 11x − 2 y = 17 x + 7 y = 13

12. ____________________________

13. Evaluate the determinant. 2 3

13. ____________________________

−6 −8 Refer to the following for problems 14–15. The sum of two numbers is 13. The difference of the same two numbers is -19. 14. Write a system of equations you could use to find the two numbers.

14. ____________________________

15. Solve the system and find the numbers.

15. ____________________________

16.

Determine the solution to the system of inequalities. 5 x − 3 y ≥ −15

16.

y

2 x + 3 y < −6 x

134

Mini-Lecture 4.1 Solving Systems of Linear Equations in Two Variables Learning Objectives: 1. 2. 3. 4.

Solve systems of linear inequalities graphically. Solve systems of linear equations by substitution. Solve systems of linear equations using the addition method. Key vocabulary: system of linear equations, solution to a system of equations, ordered triple, consistent system of equations, inconsistent system of equations, dependent system of equations, addition method

Examples: 1. Determine which, if any, of the given ordered pairs satisfy the system of linear equations. 3x − y = −1

−3x − y = 5 a) ( 2,7 )

( −1, −2 )

b)

2. Determine whether the following system of equations is consistent, inconsistent or dependent. Also indicate whether the system has exactly one solution, no solution, or an infinite number of solutions. 2x + y = 4

y = −2 x − 3 3. Solve the following system graphically. y = x −1

y = −x + 3 4. Solve each system of equations using the substitution method. a) −4 x + y = 3 b) 3x − 2 y = 5 c) 2 x + 3 y = 9

x + 2y = 6

x + 10 y = 7

3x + y = −4

5. Solve each system of equations using the addition method. x + y = 15 b) c) 2 x + 3 y = 6 a) 2 x + y = 10 4 x + 3 y = 38 4 x + 6 y = 12 2x − y = 2 Teaching Notes: • It could be pointed out that the graphing method for solving systems of equations is not always the most practical method to use, but it is still important both visually and theoretically. • You may choose to have students use the Intersect command on their graphing calculators to solve systems of linear equations graphically. Answers: 1a) no; 1b) yes; 2) inconsistent, no solution; 3) (2, 1); 4a) (0, 3); 4b) (2, 0.5); 4c) (–3, 5); 5a) (3, 4); 5b) (-7, 22); 5c) infinite number of solutions

135

Mini-Lecture 4.2 Solving Systems of Linear Equations in Three Variables Learning Objectives: 1. 2. 3. 4.

Solve systems of linear equations in three variables. Learn the geometric interpretation of a system of equations in three variables. Recognize inconsistent and dependent systems. Key vocabulary: system of three linear equations, ordered triple

Examples: 1. Solve the following system by substitution. x=5

x+ y =3 x+ y−z =2 2. Solve the following system of equations using the addition method. 3x − y + 2 z = 39 2 x − 2 y − 3 z = 18 4 x + 3 y + z = 15 3. Solve the following systems of equations using any method. b) 2a + 3b + c = −4 a) 2a + 3b + 2c = 1 3a + 2c = 10 3a + 2b + 4c = 19

a +b =1

a − b − 3c = −19

4. Determine whether the following systems are inconsistent, dependent or neither. a) a + 2b + 3c = 2 b) 2a + b + c = 3

4a + 5b + 6c = 3

a + 2b − 2c = 2

a +b + c =1

a − b + 3c = 1

Teaching Notes:

• Have students use construction paper and scissors to illustrate the various ways that three planes can intersect with each other.

Answers: 1) (5, –2, 1); 2) (8, –7, 4); 3a) (3, –2, 0.5); 3b) (–1, –3, 7); 4a) inconsistent; 4b) dependent 136

Mini-Lecture 4.3 Systems of Linear Equations: Applications and Problem Solving Learning Objectives:

1. Use systems of equations to solve applications. 2. Use linear systems in three variables to solve applications. Examples:

1. Use systems of equations in two variables to solve the following applied problems. a) There were a total of 28 students in a philosophy class. The number of men was six more than the number of women. How many men and women were in the class? b) A boat traveling with the current can go 24 miles in 2 hours. Against the current, it takes 3 hours to go the same distance. Find the rate of the boat in calm water and the rate of the current. c) Benito receives a weekly salary plus a commission, which is a percentage of his sales. One week, with sales of $2000, Benito’s gross pay was $980. On another week with sales of $3500, his gross pay was $1115. Find his weekly salary and his commission rate. d) Two trains are 330 miles apart, and their speeds differ by 20 mph. They travel toward each other and meet in 3 hours. Find the speed of each train. e) How many gallons of a 3% salt solution must be mixed with 50 gallons of a 7% solution to obtain a 5% solution? 2. Use systems of equations in three variables to solve the following applied problems. a) Melissa works at Fitz’s Pizza. Her last three orders were 5 slices of pizza, 2 salads, and 2 sodas for $9.75; 3 slices of pizza, 2 salads and 1 soda for $7.15; and 2 slices of pizza, 1 salad and 1 soda for a total of $4.35. What is the price of 1 slice of pizza, 1 salad and 1 soda sold individually? b) The sum of the measures of the three angles of a triangle is 180o. The middle-sized angle measures 2o more than 3 times the smallest angle. The middle-sized angle measures 43o less than the largest angle. Find the measure of the middle-sized angle. Teaching Notes:

• In applied problems involving money, remind students to be consistent when writing their equations; express coefficients as decimals (dollars) or whole numbers (pennies), but do not mix the two coefficient forms.

Answers: 1a) 17 men, 11 women; 1b) boat’s rate in calm water = 10 mph, current’s rate = 2 mph; 1c) weekly salary = $800, rate of commission = 9%; 1d) 65 mph and 45 mph; 1e) 50 gallons; 2a) each slice of pizza = $1.05, each salad = $1.75, each soda = $0.50; 2b) 59° 137

Mini-Lecture 4.4 Solving Systems of Equations Using Matrices Learning Objectives:

1. 2. 3. 4. 5.

Write an augmented matrix. Solve systems of linear equations. Solve systems of linear equations in three variables. Recognize inconsistent and dependent systems. Key vocabulary: matrix, matrices, elements, square matrix, augmented matrix, triangular form, row transformations

Examples:

1. Write an augmented matrix for the following system of equations.

4x − 3y = 1 5x + 2 y = 4 2. Solve the following systems of equations in two variables using matrices. a)

x + y =1 2x − y = 8

b) 3x + 2 y = 4 3x + y = 8

c)

x + 7 y = −68 4 x − 6 y = 34

3. Solve the following systems of equations in three variables using matrices. a)

x+ y+z =6 −2 x − y + z = −2 x − 2y − z = 4

b)

x+ y+ z =0 −2 x − y + z = −12 x − 2y − z =1

4. Use matrices to determine if the following systems are dependent or inconsistent. a)

3x − 2 y = 8 −3x + 2 y = −12

b)

2 x − 5 y = 19 −0.2 x + 0.5 y = −1.9

c) 2 x + y − 3z = 4 −4 x + y − z = 5 3 y − 7 z = −2

Teaching Notes:

• Using row transformations on augmented matrices is a very tedious task. Encourage students to check their arithmetic after every step. • When practicing the new procedure, also encourage students to have other students check for errors when they themselves cannot identify them.

4 −3 1 ⎤ Answers: 1) ⎡⎢ ; 2a) (3, –2); 2b) (4, –4); 2c) (–5, –9); 3a) (4, –2, 4); 3b) (2, 3, –5); ⎣ 5 2 4 ⎦⎥ 4a) inconsistent; 4b) dependent; 4c) inconsistent 138

Mini-Lecture 4.5 Solving Systems of Equations Using Determinants and Cramer’s Rule Learning Objectives:

1. 2. 3. 4. 5.

Evaluate a determinant of a 2 × 2 matrix. Use Cramer’s rule. Evaluate a determinant of a 3 × 3 matrix. Use Cramer’s rule with systems in three variables. Key vocabulary: determinant, Cramer’s rule, minor determinant, expansion by minors

Examples:

1. Evaluate the following determinants. 5 −2 3 −7 a) b) 4 3 −2 5

c)

−6 3 −1 −4

2. Solve the following systems of equations using Cramer’s Rule. a) x − 4 y = −7 3x + y = −8

b)

x+ y =3

c) 2 x + 4 y = 8 x + 2y = 6

2x − y = 5

3. Evaluate the following determinants. a)

−1 −2 −3 3 4 2 0

1

2

1 2 2

b) 2 1 0 3 3 1

c)

−1 2 1 2 1 −3 1 1 1

4. Solve the following systems of equations using Cramer’s Rule. a)

2 x + y − z = −8 4 x − y + 2 z = −3 −3 x + y + 2 z = 5

b) 3x − 2 y + 4 z = 15 x− y+z =3 x + 4 y − 5z = 0

c)

x − 4 y − 3 z = −1 −3x + 12 y + 9 z = 3 2 x − 10 y − 7 z = 5

Teaching Notes:

5 −2 ⎡ 5 −2 ⎤ • Emphasize that ⎢ does not have a numerical value but does. ⎥ 4 3 ⎣4 3 ⎦ • Indicate that Cramer’s Rule is especially useful when the solution to a system of equations involves fractional values.

8 1 Answers: 1a) 23; 1b) 1; 1c) 27; 2a) (–3, 1); 2b) ⎛⎜ , ⎞⎟ ; 2c) no solution; 3a) –3; 3b) 3; 3c) –13; 3 3 ⎝

⎠

4a) (–2, –3, 1); 4b) (3, 3, 3); 4c) infinite number of solutions 139

Mini-Lecture 4.6 Solving Systems of Linear Inequalities Learning Objectives:

1. 2. 3. 4.

Solve systems of linear inequalities. Solve linear programming problems. Solve systems of linear inequalities containing absolute value. Key vocabulary: systems of linear inequalities, linear programming, constraints

Examples:

1. Determine the solution to each system of inequalities. x≤5 a) b) 2 x + y < 4

x+ y ≥3

c) x > −2 y<4

−2 x + y > 2

2. Determine the solution to each system of inequalities. x≥0 x≥0 a) b)

y≥0 x + 2y ≤ 6

y≥0 x+ y ≤5 y ≥ 2x − 4

3. Determine the solution to each inequality. a) x ≤ 4 b) y − 3 > 2 4. Determine the solution to each system of inequalities. a) y ≤ 4 b) x − 1 > 3

y ≤ −2 x + 4

y >2

Teaching Notes:

• Students can easily confuse solving absolute value inequalities in one variable (Section 2.6) with graphing absolute value inequalities in two variables (Section 4.6). 1b)

4

−2 −2

2

4

x

−4

−2 −2

2

4

x

−4

2

4

x

2

4

−4

−2 −2

y

3b)

8

4

−2 −2 −4

2 2

4

x

−4

140

x

4

−4

−4

4

2

−4

−2 −2

2

−2 −2

6

2

y

4b) x

−4

y

−4

4

−2 −2

4

2

y

−4

2

3a)

4

−4

4a)

−2 −2

y

2b)

2 −4

2 x

−4

y 4

4

2

−4

2a)

1c)

4

2 −4

y

y

y

Answers: 1a)

2

4

4

x

2 −4

−2

2

4

x

Name:

Additional Exercises 4.1

Date:

Determine if the ordered pair or ordered triple given is a solution to each system of linear equations. 1. y = 3 x − 5 y = 4x +1 2. 4m − 3n = 6 2 m − 5n = − 4 3.

(−2, − 11)

1. ____________________________

2. ____________________________

(3, 2)

x − 3 y − z = −11

(3, − 4, 2)

3. ____________________________

2 x − y + z = 12 5 x + 4 y − z = −3

Write each equation in slope-intercept form. Without graphing the equations, state whether the system is consistent, inconsistent, or dependent. Also indicate whether the system has exactly one solution, no solution, or an infinite number of solutions.

1 x−5 2 x − 2 y = 10

4. y =

4. ____________________________

5. 4 x − y = 9

5. ____________________________

1 y = 2x − 4 2 6. 3 x − 2 y = 7 5x − y = 7 7.

6. ____________________________

Solve by graphing: x = 5y + 3 4 x − 20 y = 12

7.

y

x

8.

Solve by graphing: y = 2x + 3 y = −3 x − 7

8.

y

x

9.

Solve by graphing: x+ y =1 2x − y = 8

9.

y

x

141

Additional Exercises 4.1 (cont.) 10.

Solve by graphing: x+ y =7 3x − y = 1

Name:

10.

y

x

11. Solve the system using substitution: 3 x + 5 y = −4 y = −2 x − 5

11. ____________________________

12. Solve the system of equations by substitution: y = −8 x + 36 y = −5 x + 24

12. ____________________________

13. Solve the system using substitution: 2 x + 2 y = −5 y = −x

13. ____________________________

14. Solve the system of equations by substitution: 2x + 3 y = 5 3x + 2 y = 0

14. ____________________________

15. Solve the system using the addition method: x − y = −2 x+ y =8

15. ____________________________

16. Solve the system of equations using the addition method: x + 7 y = −68 4 x − 6 y = 34

16. ____________________________

17. Solve the system using the addition method: 4x + y = 5 2 x + 4 y = −5

17. ____________________________

18. Solve the system of equations using the addition method: 6 x+ y = 4 7 3 x− y =0 2

18. ____________________________

19. Solve using any method: y = −x + 6 1 y = (3 x + 2) 2

19. ____________________________

20. Solve using any method: 1 3 y = − x+ 4 2 y = 4x − 7

20. ____________________________

142

Name:

Additional Exercises 4.2

Date:

Solve by substitution. 1. 3x − y + z = 9 x + y + 2 z = 14

1. ____________________________

3x + 2 y − z = 5 2.

x=5 6 x + 6 y = 24 − x + 6 y − 7 z = −46

2. ____________________________

3.

x− y+ z = 6 2 x − 2 y − 2 z = 12 x + 2y − z = 0

3. ____________________________

4.

x=3 x − 2y = 7 x + y − 4 z = −3

4. ____________________________

5.

x + 2 y + z = −9

5. ____________________________

3 x + 2 y + z = −15 x − 2y − z = 3 6. 2 y + z = 7 2x − z = 3

6. ____________________________

x− y =3 Solve using the addition method. 7. 9 x − 6 y + 6 z = 1 6 x + 3 y − 3 z = −4

7. ____________________________

3x − 3 y + 6 z = 2 8.

2 x − y + 3z = 9 x + 4 y + 4z = 5 3x + 2 y + 2 z = 5

8. ____________________________

9.

3x + y − 2 z = −2 10 x + 5 y − 5 z = −6 5 x − 5 y + 10 z = −2

9. ____________________________

10.

x+ y+z =0

10. ____________________________

−2 x − y + z = −12 x − 2y − z = 1 11.

3x − y + z = 11

11. ____________________________

x + 4 y − 2 z = −12 2 x + 2 y − z = −3 12.

x + y + z = −9 −2 x − y + z = 13 x − 2 y − z = −1

12. ____________________________

143

Additional Exercises 4.2 (cont.)

Name:

Determine whether the following systems are inconsistent, dependent, or neither. 13.

2 x + y + 2 z = −5

13. ____________________________

−4 x + y + z = 2 3 y + 5z = 5 14.

2 x + y − 2 z = −1 −4 x + y − 3 z = −6 3 y − 7 z = −4

14. ____________________________

15. 2 x + y + z = 4 −x + 2 y + z = 3

15. ____________________________

9 x + 2 y + 3 z = 13 16. 2 x + y + z = 4 −x + 2 y + z = 3

16. ____________________________

7 x + 6 y + 5 z = 19 17.

2 x + y − z = −5 −4 x + y − z = 4 3 y − 3z = 3

17. ____________________________

18.

2 x + y − 3z = 4

18. ____________________________

−4 x + y − z = 5 3 y − 7 z = −2 19. 2 x + y + z = 4 −x + 2 y + z = 3

19. ____________________________

7 x + 6 y + 5 z = 19 20. 2 x + y + z = 4 −x + 2 y + z = 3

20. ____________________________

9 x + 2 y + 3 z = 13

144

Name:

Additional Exercises 4.3

Date:

1. Tickets to a local movie were sold at $5.00 for adults and $3.50 for students. If 150 tickets were sold for a total of $615.00, how many adult tickets were sold?

1. ____________________________

2. Max has cashews that sell for $4.75 a pound and peanuts that sell for $2.75 a pound. How much of each must he mix to get 80 pounds of a mixture that he can sell for $3.00 per pound?

2. ____________________________

3. A collection of quarters and dimes contains 44 coins and has a total value of $6.50. How many coins of each kind are in the collection?

3. ____________________________

4. Clare has cashews that sell for $3.50 a pound and peanuts that sell for $2.00 a pound. How much of each must she mix to obtain 60 pounds of a mixture that she can sell for $3.00 per pound?

4. ____________________________

5. Tickets for a band concert cost $8 for the main floor and $6 for the balcony. If 1125 tickets were sold and the ticket sales totaled $7700, how many tickets of each type were sold?

5. ____________________________

6. A physician invests $24,000 in two bonds. If one bond yields 6% and the other yields 12%, how much is invested in each if the annual income from both bonds is $1980?

6. ____________________________

7. An actuary invests $26,000 in two bonds. If one bond yields 12% and the other yields 7%, how much is invested in each if the annual income from both bonds is $2420?

7. ____________________________

8. An engineer invests $27,000 in two bonds. If one bond yields 7% and the other yields 9%, how much is invested in each if the annual income from both bonds is $2130?

8. ____________________________

9. Linda’s Bakery sells three kinds of cookies: chocolate chip cookies at 15 cents each, oatmeal cookies at 20 cents each, and peanut butter cookies at 25 cents each. Charles buys some of each kind and chooses three times as many peanut butter cookies as chocolate chip cookies. If he spends $4.10 on 19 cookies, how many oatmeal cookies did he buy?

9. ____________________________

10. The sum of three numbers is 161. The second number is 7 more than the first number, and the third number is 5 times the first number. Find the second number.

10. ____________________________

11. The sum of the measures of the three angles of a triangle is 180°. The middle-sized angle measures 89° less than the largest angle. The middle-sized angle measures 14° less than 3 times the measure of the smallest angle. Find the measure of the middle-sized angle.

11. ____________________________

145

Additional Exercises 4.3 (cont.)

Name:

12. The sum of three numbers is 20. The first number is the sum of second and third. The third number is three times the first. What are the three numbers?

12. ____________________________

13. Tasty Bakery sells three kinds of cookies: chocolate chip cookies at 30 cents each, oatmeal cookies at 35 cents each, and peanut butter cookies at 40 cents each. Candace buys some of each kind and chooses twice as many peanut butter cookies as chocolate chip cookies. If she spends $5.75 on 16 cookies, how many oatmeal cookies did she buy?

13. ____________________________

14. The sum of three numbers if 159. The second number is 4 more than the first number, and the third number is 3 times the first number. Find the numbers.

14. ____________________________

15. An investor has $70,000 invested in mutual funds, bonds, and a fast food franchise. She has twice as much invested in bonds as in mutual funds. Last year, the mutual funds paid a 2% dividend, the bonds paid 10%, and the fast food franchise paid 6%; her total dividend income was $4800. How much is invested in each of the three investments?

15. ____________________________

16. The sum of the measures of the three angles of a triangle is 180°. The middle-sized angle measures 8° more than 2 times the smallest angle. The middle-sized angle measures 59° less than the largest angle. Find the measure of the middle-sized angle.

16. ____________________________

17. Tasty Bakery sells three kinds of cookies: chocolate chip cookies at 20 cents each, oatmeal cookies at 25 cents each, and peanut butter cookies at 30 cents each. Melissa buys some of each kind and chooses twice as many peanut butter cookies as chocolate chip cookies. If she spends $2.85 on 11 cookies, how many peanut butter cookies did she buy?

17. ____________________________

18. The sum of three number is 286. The second number is 7 more than the first number, and the third number is 7 times the first number. Find the numbers.

18. ____________________________

19. The sum of the measures of the three angles of a triangle is 180°. The middle-sized angle measures 4° more than 4 times the smallest angle. The middle-sized angle measures 64° less than the largest angle. Find the measure of the middle-sized angle.

19. ____________________________

20. The sum of the measures of the three angles of a triangle is 180°. The middle-sized angle measures 1° more than 2 times the smallest angle. The middle-sized angle measures 38° less than the largest angle. Find the measure of the middle-sized angle.

20. ____________________________

146

Name:

Additional Exercises 4.4

Date:

Solve the system using matrices. 1. 2 x − 4 y = −16 3x + 4 y = 6

1. ____________________________

2. x − 2 y = −2 y = 4x − 6

2. ____________________________

3. 4 x + 4 y = 2 8 x + 8 y = −5

3. ____________________________

4.

2 x + 5 y = −5

4. ____________________________

4 x + 10 y = 4

5. 3 x + 2 y = 4 3x + y = 8

5. ____________________________

6. 2 x + 5 y = −15 x − 5 y = 30

6. ____________________________

7. 3 x + 4 y = 14 y = 2x − 2

7. ____________________________

8. 4 x − 3 y = 6 2 x − 5 y = −4

8. ____________________________

9. 3 x − 9 y = 12 2x − 6 y = 8

9. ____________________________

10. 4 x − 3 y = 3 x − 4y = 2

10. ____________________________

Solve the system using matrices. 11. 6 x + 4 y + 7 z = 4 − x + 9 y − 2 z = 51

11. ____________________________

5x − y + z = 1 12.

x+ y+z = 6 −2 x − y + z = −2 x − 2y − z = 4

12. ____________________________

147

Additional Exercises 4.4 (cont.) 13. −3x − 4 y + z = −19 − x − 3 y − z = −1

Name: 13. ____________________________

− x − 2 y − 3z = 7 x=7

14.

14. ____________________________

7 x + 3 y = 40 − x + 7 y − 3 z = −22 15. −3x + 4 y + 4 z = −22 3x + 3 y − 3z = 3

15. ____________________________

−4 x − 4 y − 3z = 31 16. −3x − 3 y + 4 z = −1 x − 3 y + 3z = 4

16. ____________________________

−4 x − 3 y − 3 z = 5 17. −4 x + 2 y − 2 z = 18 2 x + y + 4 z = 15

17. ____________________________

x + 3 y + 3 z = 21 18. 5 x − 2 y − 5 z = −42 − x − 8 y + 9 z = −10

18. ____________________________

3 x − y + z = −8 19. 3 x − 2 y + z = 9 6 x − 4 y + 2 z = 18

x−

19. ____________________________

2 1 y+ z =3 3 3

20. 4 x − 8 y + 6 z = 2 6 x − 12 y + 9 z = 5

20. ____________________________

3x + 8 y − 4 z = −1

148

Name:

Additional Exercises 4.5

Date:

Evaluate the determinants. 1.

2.

3.

4 3

1. ____________________________

−2 1 4

7

2. ____________________________

2 −3 1 2 −5 3 2 1 2 2 4

3. ____________________________

2 5 4

4.

4. ____________________________

4 2 3 1 1 3

5.

2 −3 2 1 3 −1 0 −2 2

5. ____________________________

6.

3 3 1 4 3 2 1 1 2

6. ____________________________

7.

1 −7 −1 6 2 −2 9 4 −9

7. ____________________________

8.

9.

7 12

8. ____________________________

7 12 −1 4

9. ____________________________

−2 6

1 4 −3 10. 0 −2 8 3 6 0

10. ____________________________

149

Additional Exercises 4.5 (cont.)

Name:

Solve the system of equations using determinants. 11.

x − 5 y = −5

11. ____________________________

4 x + 5 y = −4

12. 5 x + 3 y = −5 5 x + y = −1

12. ____________________________

13. 8 x + y = 36 5 x + y = 21

13. ____________________________

14. x = 3 y + 4 −2 x + 6 y = −8

14. ____________________________

15.

x + 3y − z = 8

15. ____________________________

3 x + 3 y + 3 z = −6 2 x + 3 y + 3 z = −3

16.

x + 2y + z = 5

16. ____________________________

x+ y−z =6 5x + 8 y + z = 7

17.

x + 2y + z = 8 x+

17. ____________________________

y−z =9

9 x + 14 y + z = 10 18. 2 x + y − 3 z = −2 x − 2 y − 3 z = −13

18. ____________________________

3 x − 2 y − z = −3 19. 2 x − y − 3z = 17 3 x + y + z = −2

19. ____________________________

x + y + 2 z = −9 20. 2 x − y + 3 z = 2 x + 2 y − 3 z = −10

20. ____________________________

3x − 3 y − 3 z = −21

150

Name:

Additional Exercises 4.6

Date:

Determine the solution to each system of inequalities. 1.

y≥x+7 3x + y ≤ 3

1.

y

x

2.

2 x + y ≥ −2

2.

y

6x + 3y ≤ 6 x

3.

y ≤ − 12 x + 3

3.

y

y ≥ x−3 x

4.

y > 2x − 6 x+ y <0

4.

y

x

5.

2x + 3y ≤ 6

5.

y

x − 3 y ≥ −9 x

6.

y ≥ –2x – 2 y≤x+2

6.

y

x

151

Additional Exercises 4.6 (cont.) 7.

x − 3y > 6

Name:

7.

y

2x + y > 5 x

8.

y ≥ 2x + 3 y ≤ –x + 4

8.

y

x

9.

2x + 5y ≥ 10 x≤y x≤5

9.

y

x

10.

x≥0 y≥0

10.

y

x + y ≤ 10 3 x + y ≤ 18

x

x≥0 y≥0

11.

11.

y

3 x + 5 y ≤ 19 x + 4 y ≤ 11

12.

x

2x + 3 y ≥ 6

12.

y

x≥ y x≤8 x

152

Additional Exercises 4.6 (cont.) x≥0 y≥0

13.

Name:

13.

y

x+ y ≤9 2 x + y ≤ 15

x

x≥0 y≥0

14.

14.

y

4 x + 3 y ≤ 23 x + 3 y ≤ 17

15.

x

3 x + 5 y ≥ 15

15.

y

x≤ y x≤6 x

16.

x≥0 y≥0

16.

y

x + y ≤ 11 3x + y ≤ 21

17.

x

x−4 ≥1

17.

y

x+ y ≤ 4 x

18.

x−2 ≥3

18.

y

x+ y ≤ 2 x

153

Additional Exercises 4.6 (cont.) 19.

x <4

Name:

19.

y

y ≤4 x

20.

x <6

20.

y

y ≥3 x

154

Name:

Chapter 4 Test Form A

Date:

Without graphing or solving the system of equations, identify the type of system as consistent, dependent, or inconsistent. State whether the system has exactly one solution, no solution, or an infinite number of solutions. 1. 3 x − y = 7 y = 3x + 9 2.

1. ____________________________

x + 5y = 2 2 x + 10 y = 4

2. ____________________________

3. y − x = 1 y+x=3

3. ____________________________

Solve the system of equations by the method indicated. 4.

y = x +1 y+x=3

graphically

4.

y

x

5.

y = −x + 4 y = x−2

graphically

5.

y

x

6. x + y = 5 x = y +1

substitution

7. 5 x + 4 y = −26 y = 3x + 2 8. 2 x − y = 3 3x + y = 2 9.

x+ y =3

6. ____________________________

substitution

7. ____________________________

addition

8. ____________________________

addition

9. ____________________________

2x − 3y = 1

10. 5 x − 3 y = 12 3 x + 5 y = 14

addition

10. ____________________________

11. Write the augmented matrix for the system of equations. x− y+ z = 6

2x + 3y + 2z = 2 5 y + 4 z = −2

155

11. ____________________________

Chapter 4 Test Form A (cont.) 12. Consider the augmented matrix below. Show the results obtained by multiplying the elements in the first row by –1 and adding the products to their corresponding elements in the second row.

Name: 12. ____________________________

⎡ 1 1 1 4⎤ ⎢ 1 −2 −1 1⎥ ⎢ ⎥ ⎢⎣ 2 −1 −2 −1⎥⎦

13. Solve the system of equations using matrices. x+ y =9 2 x − y = −3

13. ____________________________

Evaluate the following determinants. 14.

5 −2 −1

14. ____________________________

7

0 2 0 15. 3 −1 1 1 −2 2

15. ____________________________

16. Solve the system using determinants and Cramer’s Rule. 3x − y = 4 7x + 2 y = 5

16. ____________________________

Express the problem as a system of linear equations and use the method of your choice to find the solution to the problem. 17. A mechanic has 2% and 6% solutions of alcohol. How much of each solution should she mix to obtain 60 liters of a 3.2 solution?

17. ____________________________

18. The sum of three numbers is 5. The first number minus the second plus the third is 1. The first minus the third is 3 more than the second. Find the numbers.

18. ____________________________

Graph the system of inequalities and indicate its solution. 19.

y ≤ −2 x y≥x

19.

y

x

20.

y >1

20.

y

x < −2 y > x+5 x

156

Name:

Chapter 4 Test Form B

Date:

Without graphing or solving the system of equations, identify the type of system as consistent, dependent, or inconsistent. State whether the system has exactly one solution, no solution, or an infinite number of solutions. 1. 3 y = x − 2 3x − 9 y = 6

1. ____________________________

3x − y = 2

2. ____________________________

2.

6x − 2 y = 3

3. x − y = 2 y = −x + 4

3. ____________________________

Solve the system of equations by the method indicated. 4.

1 x −1 2 x+ y =5 y=

4.

graphically

y

x

5.

2x − 3 y = 6

graphically

5.

y

x+ y =3

x

6. y = 1 + x x + 3 y = −1 7. 2 y + x = 1 y − 2x = 8

substitution

6. ____________________________

substitution

8. 4 x − 3 y = 1 5 x + 3 y = −10 9. 3 x + y = 20 x + y = 12 10. 2 x − 3 y = −11 3 x + 2 y = 29

7. ____________________________

addition

8. ____________________________

addition

9. ____________________________

addition

10. ____________________________

11. Write the augmented matrix for the system of equations. 2x − 3y + z = 5

11. ____________________________

x + 3 y + 8 z = 22

3 x − y + 2 z = 12

157

Chapter 4 Test Form B (cont.) 12. Consider the augmented matrix below. Show the results obtained by multiplying the elements in the first row by –1 and adding the products to their corresponding elements in the second row.

Name: 12. ____________________________

⎡ 1 1 1 2⎤ ⎢ 1 −2 −1 2 ⎥ ⎢ ⎥ ⎢⎣3 2 1 2 ⎥⎦

13. Solve the system of equations using matrices. x+ y =5 2 x + 6 y = 22

13. ____________________________

Evaluate the following determinants. 14.

15.

2 −5 6

14. ____________________________

7

1 2 1 2 −2 −2 1 6 3

15. ____________________________

16. Solve the system using determinants and Cramer’s Rule. 2x − y = 6 x + 3y = 4

16. ____________________________

Express the problem as a system of linear equations and use the method of your choice to find the solution to the problem. 17. The sum of three numbers is 57. The second is 3 more than twice the first. The third is 6 more than the first. Find the numbers.

17. ____________________________

18. A pharmacy has 5% and 25% saline solutions. How much of each must be mixed to obtain 6 liters of a 20% solution?

18. ____________________________

Graph the system of inequalities and indicate its solution. 19.

y > 2x − 3 y < −2 x + 3

19.

y

x

20.

x ≥1

20.

y

y≤2

2x + 3 y ≥ 6 x

158

Name:

Chapter 4 Test Form C

Date:

Without graphing or solving the system of equations, identify the type of system as consistent, dependent, or inconsistent. State whether the system has exactly one solution, no solution, or an infinite number of solutions. 1. y + x = 5 y = x−3

1. ____________________________

2. x + 2 y = 6 2x = 8 − 4 y

2. ____________________________

x−3 = y

3. ____________________________

3.

2x − 2 y = 6

Solve the system of equations by the method indicated. 4.

x+y=1 y=x–3

graphically

4.

y

x

5.

y=3 graphically x + y = –1

5.

y

x

6. 3 x − 8 y = 32 x = 8 y + 16 7. x = 3 y + 2 3x − 9 y = 6

substitution

substitution

7. ____________________________

addition

8. ____________________________

addition

9. ____________________________

8. 3x + 2 y = 20 5 x − 2 y = −4 9.

6. ____________________________

x + 3y = 2 3 x + 5 y = −2

10. 5 x + 8 y = −11 7 x − 6 y = 19

addition

10. ____________________________

11. Write the augmented matrix for the system of equations. 6 x − 4 y + 5 z = 31

11. ____________________________

5 x + 2 y + 2 z = 13 x+ y+ z = 2

159

Chapter 4 Test Form C (cont.) 12. Consider the augmented matrix below. Show the results obtained by multiplying the elements in the second row by 3 and adding the products to their corresponding elements in the third row.

Name: 12. ____________________________

⎡ 1 1 1 6⎤ ⎢0 1 1 5⎥⎥ ⎢ ⎢⎣ 0 −3 1 −3⎥⎦

13. Solve the system of equations using matrices. x + 2y = 0 2 x + 5 y = −1

13. ____________________________

Evaluate the following determinants. 14.

−4 5

14. ____________________________

1 7

3 4 0 15. 0 1 2 1 −1 5

15. ____________________________

16. Solve the system using determinants and Cramer’s Rule. 3x − 2 y = 7 3x + 2 y = 9

16. ____________________________

Express the problem as a system of linear equations and use the method of your choice to find the solution to the problem. 17. A 150-foot rope is cut into two pieces. If one piece is 2 feet more than three times the other piece, find the length of the two pieces.

17. ____________________________

18. Ann invested $9000 in two accounts. One account gave 10% interest and the other 8%. Find the amount placed in each account if she received $840 in interest after one year.

18. ____________________________

Graph the system of inequalities and indicate its solution. 19.

y < x−3 y > −2 x

19.

y

x

20.

y≤3

20.

y

x ≥ −4 y≥

1 x+2 2

x

160

Name:

Chapter 4 Test Form D

Date:

Without graphing or solving the system of equations, identify the type of system as consistent, dependent, or inconsistent. State whether the system has exactly one solution, no solution, or an infinite number of solutions. 1. 3 x = 6 + 9 y 3y = x − 2

1. ____________________________

2. x + y = 5 y = x−3

2. ____________________________

3. y = 2 x + 1 2x − 3 = y

3. ____________________________

Solve the system of equations by the method indicated. 4.

y=x–3 y = –2x

graphically

4.

y

x

5.

2 x−4 3 x + y = –4 y=

5.

graphically

y

x

6. x − y = 2 y = −x + 4

substitution

6. ____________________________

7. y = 2 x − 8 −3 x + y = − 2

substitution

7. ____________________________

8. 3 x + 5 y = 10 x − 5 y = −10

addition

8. ____________________________

9. 5 x − 3 y = −2 − x + 2 y = −8

addition

9. ____________________________

10. 2 x − 4 y = 0 3 x + 5 y = 11

addition

10. ____________________________

11. Write the augmented matrix for the system of equations. 3x − 4 y + z = 7

5 x + y − 3z = 11 7 y + 2 z = −5

161

11. ____________________________

Chapter 4 Test Form D (cont.) 12. Consider the augmented matrix below. Show the results obtained by multiplying the elements in the second row by –5 and adding the products to their corresponding elements in the third row.

Name: 12. ____________________________

⎡ 1 −2 −1 −2 ⎤ ⎢5 1 0 6 ⎥⎥ ⎢ ⎢⎣ 0 3 1 4 ⎥⎦

13. Solve the system of equations using matrices. x + y = 12 3 x + y = 20

13. ____________________________

Evaluate the following determinants. 14.

−7 6

14. ____________________________

−3 2

1 0 1 15. 1 3 0 1 −2 4

15. ____________________________

16. Solve the system using determinants and Cramer’s Rule. x − y = −2 x + 2 y = 10

16. ____________________________

Express the problem as a system of linear equations and use the method of your choice to find the solution to the problem. 17. A collection of nickels and dimes has a value of $3.70. If there are a total of 52 coins, how many nickels and dimes are there?

17. ____________________________

18. A gardener has 5% and 15% solutions of fertilizer. How much of each should he mix to obtain 100 liters of a 12% solution?

18. ____________________________

Graph the system of inequalities and indicate its solution. 19.

1 y < − x +1 2 y>x

19.

y

x

20.

y ≥ −2

20.

y

x ≥1 y ≤ −x + 4 x

162

Name:

Chapter 4 Test Form E

Date:

Without graphing or solving the system of equations, identify the type of system as consistent, dependent, or inconsistent. State whether the system has exactly one solution, no solution, or an infinite number of solutions. 1. 2 x + 3 y = 6 4 x + 6 y = −12

1. ____________________________

2. 2 x + y = 4 x− y =5

2. ____________________________

3. x − y = 3 y = x−3

3. ____________________________

Solve the system of equations by the method indicated. 4.

1 y = − x +1 2 y = x−5

graphically

4.

y

x

5.

y = 3x + 2 y=x+2

graphically

5.

y

x

6. 3 x + y = 5 4x + 5 y = 3

substitution

6. ____________________________

7. 2 x = 8 − 4 y x = 6 − 2y

substitution

7. ____________________________

8. 2 x + 6 y = 0 −2 x − 5 y = 0

addition

8. ____________________________

9. 3 x − 5 y = 7 x − 2y = 3

addition

9. ____________________________

10. 3 x + 5 y = 13 5 x − 3 y = −1

addition

10. ____________________________

11. Write the augmented matrix for the system of equations. −2 x − y − z = −3

3 x − 2 y − 2 z = −5 −x + y

=0

163

11. ____________________________

Chapter 4 Test Form E (cont.) 12. Consider the augmented matrix below. Show the results obtained by multiplying the elements in the second row by –6 and adding the products to their corresponding elements in the third row.

Name: 12. ____________________________

⎡ 1 3 −2 2 ⎤ ⎢ 0 1 4 5⎥ ⎢ ⎥ ⎢⎣ 0 6 −1 5⎥⎦

13. Solve the system of equations using matrices. x+ y =3 2x − 3 y = 6

13. ____________________________

Evaluate the following determinants. 14.

7 −2 9

14. ____________________________

2

1 −3 7 15. 1 1 1 1 −2 3

15. ____________________________

16. Solve the system using determinants and Cramer’s Rule. x + y = −4 2 x − 3 y = 12

Express the problem as a system of linear equations and use the method of your choice to find the solution to the problem. 17. A boat can travel 20 miles per hour with the current and 12 miles per hour against the current. Find the speed of the current and the speed of the boat in still water. 18. The sum of three numbers is 26. Twice the first minus the second is 2 less than the third. The third is the second minus 3 times the first. Find the numbers.

16. ____________________________

17. ____________________________

18. ____________________________

Graph the system of inequalities and indicate its solution. 19.

y

y < x +1 y > x −1

19.

x

20.

y

x ≥1

20.

y ≥1 x+ y ≤ 4 x

164

Name:

Chapter 4 Test Form F

Date:

Without graphing or solving the system of equations, identify the type of system as consistent, dependent, or inconsistent. State whether the system has exactly one solution, no solution, or an infinite number of solutions. 1. x + y = 10 y = −x + 5

1. ____________________________

2. x + 4 y = 2 2x − y = 1

2. ____________________________

3. x − 3 y = 2 9 y = 3x − 6

3. ____________________________

Solve the system of equations by the method indicated. 4.

y = −2 x y = x+3

y

graphically

4.

x

5.

y = −3 x + y = −2

y

graphically

5.

x

6. x + y = 30 x = y+6

substitution

6. ____________________________

7. x = 3 y − 5 −x − 3y = 1

substitution

7. ____________________________

addition

8. ____________________________

addition

9. ____________________________

addition

10. ____________________________

8. 2 x − 3 y = −6 2x − y = 2 9.

4x − y = 6 3 x + 2 y = 21

10. 5 x + 7 y = −1 4 x − 2 y = 22

11. Write the augmented matrix for the system of equations. x+ y =6

x− y+z =3 x

−z=5

165

11. ____________________________

Chapter 4 Test Form F (cont.) 12. Consider the augmented matrix below. Show the results obtained by multiplying the elements in the first row by –4 and adding the products to their corresponding elements in the third row.

Name: 12. ____________________________

⎡1 5 0 6⎤ ⎢0 2 1 3⎥ ⎢ ⎥ ⎢⎣ 4 0 3 7 ⎥⎦

13. Solve the system of equations using matrices. 5 x + 2 y = −23 2 x + y = −1 0

13. ____________________________

Evaluate the following determinants. 14.

15.

2 8

14. ____________________________

−1 7 1 −2 3 3 1 1 2 −1 −2

15. ____________________________

16. Solve the system using matrices and Cramer’s Rule. 3x − y = 4 −2 x + 5 y = −1

16. ____________________________

Express the problem as a system of linear equations and use the method of your choice to find the solution to the problem. 17. John picked a total of 87 quarts of strawberries in three days. Tuesday’s yield was 15 quarts more than Monday’s. Wednesday’s yield was 3 less than Tuesday’s. How many quarts did he pick on Monday?

17. ____________________________

18. Admission for a school play was $10 for adults and $5 for children. A total of 300 tickets were sold. How many adults tickets were sold if a total of $2500 was collected?

18. ____________________________

Graph the system of inequalities and indicate its solution. 19.

y

y < 2x − 4 x+ y > 4

19.

x

20.

y

x ≥1

20.

y≤3 y ≥ −x x

166

Name:

Chapter 4 Test Form G

Date:

Without graphing or solving the system, use the following systems of equations to answer questions 1–3. A. x + y = 9 B. x = 3 y + 1 C. y = 5 x + 2 y = 2x + 3 6 y = 2x + 5 10 x − 2 y = −4 1. For which system is there exactly one solution? (a) A (b) B

(c) C

(d) none of these

2. Which system is inconsistent? (a) A (b) B

(c) C

(d) none of these

3. Which system is dependent with an infinite number of solutions? (a) A (b) B (c) C

(d) none of these

Solve the system of equations by the method indicated. 4. y = − x + 4 graphically y = x−2 y

(a)

y

(b)

4

4

2 (—3, —1)—2 0 —2

2

x

4

—4 —2 0 —2

—4

4

2 —4 —2 0 —2

2

2

(3, 1) x 4

x —4 —2 0 —2

—4

4 x 3 2 y = x−2 3 (a)

2

(3, —1)

—4

graphically

y

(b)

(—3, —4)

4

2

2 x

4

—2 0

—4

(3, 4)

2

4

x

—4

y

(d)

4 (4, 3) 2 —4 —2 0 —2

y

4

—4 —2 0

(c)

x

4

y

(d)

4

5. y =

2

—4

y

(c)

2

(—3, 1)

2

4

y 4 2

x

—4 —2 0 (—4, —3)

—4

—4

167

2

x

Chapter 4 Test Form G (cont.) 6. 2x + y = 3 y = 3x − 7

substitution

(a) (2, –1) 7. y − x = −3 y = x −3

(d) (2, 1)

(b) (3, 0)

(c) no solution

(d) infinite number

(b) (–9, 1)

⎛1 ⎞ (c) ⎜ , − 1⎟ ⎝9 ⎠

(d) (9, –1)

(b) (–1, 3)

(c) (3, –1)

(d) (–1, –3)

(b) (1, 6)

(c) (–1, –6)

(d) (1, –6)

⎡1 −3 1 −2 ⎤ (c) ⎢⎢1 1 1 3⎥⎥ ⎢⎣1 −2 3 1⎥⎦

⎡ 1 −3 0 −2 ⎤ (d) ⎢⎢ 1 1 1 3⎥⎥ ⎢⎣ 0 −2 3 1⎥⎦

addition

(a) (1, 3) 10. 7 x + 3 y = 25 2x + y = 8

(c) (1, 2)

addition

⎛ 1 ⎞ (a) ⎜ − , 1⎟ ⎝ 9 ⎠

9. 3 x + 2 y = 7 2x − y = 7

(b) (–1, 2)

substitution

(a) (0, –3) 8. 9 x + 3 y = 2 −9 x − y = 0

Name:

addition

(a) (–1, 6)

11. Write the augmented matrix for the system of equations. x − 3y = −2

x+ y+ z =3

−2 y + 3z = 1 ⎡1 −3 0 2 ⎤ (a) ⎢⎢1 1 1 3⎥⎥ ⎢⎣1 −2 3 1⎥⎦

⎡ 0 −3 0 −2 ⎤ (b) ⎢⎢ 0 0 0 3⎥⎥ ⎢⎣ 0 −2 3 1⎥⎦

12. Consider the augmented matrix. Determine the results obtained by multiplying the elements in the second row by –5 and adding the products to their corresponding elements in the third row. ⎡ 1 0 4 5⎤ ⎢0 1 3 4⎥ ⎢ ⎥ ⎢⎣ 0 5 −2 3⎥⎦

4 5⎤ ⎡1 0 (a) ⎢⎢ 0 1 3 4 ⎥⎥ ⎢⎣ 0 −5 −15 −20 ⎥⎦

4 5⎤ ⎡1 0 (b) ⎢⎢ 0 −5 −15 −20 ⎥⎥ ⎢⎣ 0 5 −2 3⎥⎦

4 5⎤ ⎡1 0 (c) ⎢⎢ 0 1 3 4 ⎥⎥ ⎢⎣ 0 0 −17 −17 ⎥⎦

4 5⎤ ⎡1 0 (d) ⎢⎢ 0 0 −17 −17 ⎥⎥ ⎢⎣ 0 5 −2 3⎥⎦

(c) (7, 1)

(d) (4, 7)

13. Solve the system of equations using matrices. 3x + 2 y = 2 x+ y =3 (a) (3, –1)

(b) (–4, 7)

168

Chapter 4 Test Form G (cont.)

Name:

Evaluate the following determinants. 14.

6 −9 2

3

(a) 0

15.

(b) 36

(c) –36

(d) –15

(b) –13

(c) 1

(d) 3

1⎞ ⎛1 (c) ⎜ , − ⎟ 2 3⎠ ⎝

⎛ 1 1⎞ (d) ⎜ − , − ⎟ 3⎠ ⎝ 2

1 −1 2 2 1 −2 1 −2 6

(a) 6

16. Solve the system using matrices and Cramer’s Rule. 2x − 3y = 0 −4 x + 3 y = −1 ⎛ 1 1⎞ (a) ⎜ − , ⎟ ⎝ 2 3⎠

⎛1 1⎞ (b) ⎜ , ⎟ ⎝ 2 3⎠

Express the problem as a system of linear equations and use the method of your choice to find the solution. 17. The sum of three numbers is 6. Twice the first plus 3 times the second is 9. The third number is one less than the first. Find the numbers. (a) 3, 1, 2 (b) –1, 4, 3 (c) 2, 3, 1 (d) 5, –3, 4 18. Chad sold 30 decorated sweatshirts at a craft show. White ones cost $10.00 and black ones cost $15.00. He received a total of $390 for the shirts. How many black shirts did he sell? (a) 24 (b) 20 (c) 18 (d) 15 19. Graph the system of inequalities and indicate its solution. y < x+3 3 y > − x+3 4 y y (a) (b (c ) ) 4 4 —2 0 —2

2

4

x

—2 0 —2

—4

2

x

4

y 4

—2 0 —2

—4

y

(d )

2

4

x

4

—2 0 —2

—4

—4

20. Identify the system of equalities shown by the graph y 4

—4 —2 0

(a) x ≤ −1 y≤2 y≤x

2

4

x

(b) x ≤ −1 y≤2 y≥x

(c) y ≤ −1 y≤2 y≤x

169

(d) y ≤ −1 x≤2 y≥x

2

4

x

Name:

Chapter 4 Test Form H

Date:

Without graphing or solving the system, use the following systems of equations to answer questions 1–3. 1 A. x + y = 5 B. y = x − 2 C. y = x − 4 3 x = y+5 x − 3y = 6 x − y = −4 1. For which system is there exactly one solution? (a) A (b) B

(c) C

(d) none of these

2. Which system is inconsistent? (a) A (b) B

(c) C

(d) none of these

3. Which system is dependent with an infinite number of solutions? (a) A (b) B (c) C

(d) none of these

Solve the system of equations by the method indicated. 4. y = x − 3 y = −x + 5

graphically y

(a)

(b)

y

4 2

2

(4, 1) x 4

—4 —2 0 —2

x —4 —2 0 —2

2

(4, —1)

—4 y

(c)

y

(d)

4 2 —4

2

0

2

0 (—4, —1) —2 —2

(1, —4)

—4

5. y = − x x=3 (a)

x

2

4

—4

graphically y

(b)

y

4

4

(—3, 3) 2

2

(3, 3) —4 —2 0 —2

2

x

4

—4 —2 0 —2

—4

(c)

2

4

x

2

4

x

—4

y

(d)

y

4

4

2

2

—4 —2 0 (—3, —3)—2

2

4

x

–4 –2 0 –2

—4

–4

170

(3, –3)

x

Chapter 4 Test Form H (cont.) 6. 2 y + 3 x = 21 y = 4x − 6

substitution

(a) (–3, –6)

(b) (–3, 6)

7. 2 x − 3 y = 10 y=

(c) (3, –6)

(d) (3, 6)

5⎞ ⎛ (b) ⎜ −1, ⎟ 2⎠ ⎝

(c) no solution

(d) infinite number

(b) (2, –6)

(c) (–2, 6)

(d) (2, 6)

⎛ 1⎞ (c) ⎜ 4, ⎟ ⎝ 3⎠

1⎞ ⎛ (d) ⎜ −4, ⎟ 3⎠ ⎝

(c) (0, –1)

(d) (–1, 0)

⎡ 1 0 −3 6 ⎤ (c) ⎢⎢ 0 1 2 2 ⎥⎥ ⎢⎣ 7 −3 0 9 ⎥⎦

⎡ 0 0 −3 6 ⎤ (d) ⎢⎢ 0 0 2 2 ⎥⎥ ⎢⎣ 7 −3 0 9 ⎥⎦

substitution

2 x 3

(a) (3, –8) 8. 3 x + 7 y = 48 5 x − 7 y = −32 (a) (–2, –6) 9. −2 x + 3 y = −9 x + 3y = 3 1⎞ ⎛ (a) ⎜ 4, − ⎟ 3⎠ ⎝

10.

Name:

addition

addition 1⎞ ⎛ (b) ⎜ −4, − ⎟ 3⎠ ⎝

6 x + 3 y = −6 −4 x + 7 y = 4

(a) (0, 1)

addition (b) (1, 0)

11. Write the augmented matrix for the system of equations. a − 3c = 6

b + 2c = 2

7a − 3b

=9

⎡ 1 −3 6 ⎤ (a) ⎢⎢ 1 2 2 ⎥⎥ ⎢⎣ 7 −3 9 ⎥⎦

⎡ 1 1 −3 6 ⎤ (b) ⎢⎢ 1 1 2 2 ⎥⎥ ⎢⎣ 7 −3 1 9 ⎥⎦

12. Consider the augmented matrix. Show the results obtained by multiplying the elements in the first row by –5 and adding the products to the corresponding elements in the second row. ⎡ 1 −1 2 2 ⎤ ⎢ 5 0 −3 2 ⎥ ⎢ ⎥ ⎢⎣ 0 −4 6 2 ⎥⎦ 2 2⎤ ⎡ −5 5 −10 −10 ⎤ ⎡ 1 −1 ⎢ ⎢ ⎥ (a) ⎢ 5 0 −3 2 ⎥ (b) ⎢ 0 5 −13 −8⎥⎥ 6 2 ⎥⎦ 6 2 ⎦⎥ ⎣⎢ 0 −4 ⎣⎢ 0 −4

⎡ 0 5 −15 −8⎤ (c) ⎢⎢ 5 0 −3 2 ⎥⎥ 6 2 ⎥⎦ ⎣⎢ 0 −4

2 2⎤ ⎡ 1 −1 ⎢ (d) ⎢ 5 0 −3 2 ⎥⎥ ⎣⎢ 0 5 −15 −8⎦⎥

(c) (1, 2)

(d) (–1, –2)

13. Solve the system of equations using matrices. x + 5 y = −1 1 2 x + y = −4 (a) (1, –2)

(b) (–1, 2)

171

Chapter 4 Test Form H (cont.)

Name:

Evaluate the following determinants. 14.

3 −2 3

2

(a) 0

(b) –12

(c) 12

(d) 5

(b) 3

(c) 9

(d) –5

(c) (3, 2)

(d) (–2, –3)

−1 −2 −3

15.

3 0

4 1

2 2

(a) –3

16. Solve the system using matrices and Cramer’s Rule. 2 x + 3 y = 12 −4 x + 5 y = −2 (a) (–2, 3)

(b) (–3, 2)

Express the problem as a system of linear equations and use the method of your choice to find the solution. 17. The sum of the angles in a triangle is 180°. In triangle ABC, the measure of angle B is 2° more than three times the measure of angle A. The measure of angle C is 8° more than the measure of angle A. Find the measure of angle A. (a) 34° (b) 42° (c) 104° (d) 114° 18. Two investments are made totaling $8800. In one year, these investments yield $663 in simple interest. Part of the $8800 was invested at 7% and part at 8%. Find the amount invested at 7%. (a) $2,700 (b) $3,000 (c) $5,000 (d) $4,100 19. Graph the system of inequalities and indicate its solution. x − y > −2 y < −x y

(a)

(b )

4 2

y

(c )

4 2

—4 —2 0 —2

2

4

—4

x

—4 —2 0 —2

y

y

(d )

4

4

2 2

x

4

—4

—4 —2 0 —2 —4

2 2

4

x

—4 —2 0 —2 —4

20. Identify the system of equalities shown by the graph y

2 —4 —2 0

2

(a) x ≥ 1 y ≤ −1 y ≤ x−5

4

x

(b) x ≥ 1 y ≤ −1 y ≥ x−5

(c) y ≥ 1 x ≤ −1 y ≤ x+5

172

(d) y ≥ 1 x ≤ −1 y ≥ x+5

2

4

x

Cumulative Review Test 1–4 Form A

Name: Date:

1. Evaluate: 62 + 9 ÷ 3 − 2 ⋅ 5 .

1. ____________________________

2. Consider the set of numbers: 1 1 ⎧ ⎫ ⎨−5, 25, − 5, 0, 3, π, , − , 1.75⎬ 2 5 ⎩ ⎭ List those that are rational.

2. ____________________________

3. List from largest to smallest: −2, 7 , − (−4), − −3 .

3. ____________________________

Solve. 4. 3 ( x − 3) = 2 ( 6 − x )

4. ____________________________

x 3x + =7 4 5

5. ____________________________

6. 5 x + 2 = 5

6. ____________________________

7. Solve A = πr 2 for π.

7. ____________________________

8. Find the solution set of the inequality. −2 < 5 − 2 x ≤ 6

8. ____________________________

5.

⎛ 5x 5 y −1 ⎞ 9. Simplify: ⎜ 3 ⎟ ⎝ xy ⎠

−2

9. ____________________________

10. Graph 6 x − 5 y = 30 .

10.

y

x

11. Write the slope-intercept form of the equation of the line that is perpendicular to the line 2 x + 4 y = 5 and passes through the point (2, –3).

173

11. ____________________________

Cumulative Review Test 1–4 Form A (cont.) 12. Graph the inequality 7 x + 5 y ≥ −35 .

Name:

12.

y

x

13. Determine whether or not the graph shown is a function.

13. ____________________________

y 4 2 —4 —2 0 —2

2

4

x

—4

14. If f ( x) = x 2 + 5 x + 7 , find f (−3) .

14. ____________________________

Solve each system of equations. 15. y = 2 − 2 x 5 x + 4 y = −1 16.

15. ____________________________

3x − y = 5

16. ____________________________

−4 x + y = −5

17. 2 x − y = −11 2 x + z = −6

17. ____________________________

3y − z = 7 18. If two angles in a triangle are the same size and the third angle is 30° less, find the measure of the three angles.

18. ____________________________

19. How many pounds of coffee beans selling for $2.60 per pound should be mixed with 2 pounds of coffee beans selling for $1.80 a pound to obtain a mixture selling for $2.44 per pound?

19. ____________________________

20. Tickets to a local movie were sold at $4.00 for adults and $2.50 for students. If 270 tickets were sold for a total of $825.00, how many adult tickets were sold?

20. ____________________________

174

Name:

Cumulative Review Test 1–4 Form B

Date:

1. Evaluate the expression: 7 − 5(2 − 6) 2 (a) 32

(b) –73

(c) –14

(d) 87

1⎫ ⎧ 2. Choose the elements of that set ⎨−0.5, 13, 0, − 13, − 6, 5, ⎬ that are rational numbers. 2⎭ ⎩

(b) −0.5, 13, 0, − 6,

(a) 13

1 2

(d) − 13, 5

(c) 13, 0, –6

3. Which of the following lists values from smallest to largest? (a) −18 , 14, −5 , 0

(b) −18 , −5 , 0, 14

(

(c) 0, −5 , 14, −18

(d) −5 , −18 , 0, 14

)

4. Solve: 18a = 5 12a − ⎡⎣ 2a + (15a − 9 ) ⎤⎦ + 11 (a) a = −

34 43

5. Solve for x:

(b) a = −

56 43

(c) a =

56 43

(d) a =

34 43

x −8 x = 6 7

(a) 56

(b) 7

(c) –56

(d) –7

6. Solve 2 x − 5 = 13 . (a) {9}

(b)

7. Solve the formula m =

{−4}

(c)

(d)

{4, − 9}

x − x1 for x. 8−3

(a) x = 5m − x1

(b) x = 5m + x1

8. Find the solution set: −2 ≤

7 2⎫ ⎧ (a) ⎨ x − ≤ x < − ⎬ 5 5 ⎩ ⎭

{−4, 9}

(c) x = m + 5 x1

(d) x = m − 5 x1

5x + 1 < 1. 3

⎧ 2 (b) ⎨ x − ≤ x < 5 ⎩

7⎫ ⎬ 5⎭

⎧ 2 (c) ⎨ x ≤ x < ⎩ 5

7⎫ ⎬ 5⎭

7 ⎧ (d) ⎨ x − ≤ x < 5 ⎩

−2

⎛ 3 x3 y −4 ⎞ 9. Simplify ⎜ ⎟ . ⎝ 6y ⎠

(a)

y10 4 x6

(b)

y6 4 x6

(c)

175

4x 6 y10

(d)

4x 6 y6

2⎫ ⎬ 5⎭

Cumulative Review Test 1–4 Form B (cont.)

Name:

10. Which equation is graphed below? y 4 2 —4 —2 0 —2

2

x

4

—4

(a) 3 x + 4 y = 12

(b) 3 x − 4 y = 12

(c) 3 x − 4 y = −12

(d) 3 x + 4 y = −12

11. Which is the equation of the line that is perpendicular to the line x − 2 y = 6 and passes through the point (3, – 1)? 1 1 5 (a) y = x + 5 (b) y = x − (c) y = 2 x − 7 (d) y = −2 x + 5 2 2 2 12. Which inequality is graphed below? y 4 2 –4 –2 0 –2

2

x

4

–4 –6

(a) 7 x − 4 y < 28

(b) 7 x − 4 y ≤ 28

(c) 7 x + 4 y < 28

13. Determine which of the following graphs are functions. y y (a) (b)

(d) 7 x + 4 y ≤ 28

y

(c)

y

(d)

4

4

6

4

2

2

4

2

—4 —2 0 —2

2

—4 —2 0 —2

2

4

x

2

4

x

—4 —2 0 —2

—4

—4

2

4

–4 –2 0 –2

x

–4

14. If h( x) = 3 x 2 − 5 x + 7 , find h(−1) . (a) 5

(b) 6

(c) 15

(d) –1

(c) (–2, –5)

(d) (–1, –4)

15. Solve the system: 3 x − 2 y = 4 y = x−3 (a) no solution

19 ⎞ ⎛ (b) ⎜ −5, − ⎟ 2⎠ ⎝

16. Solve the system: 2 x − y = 5 3x + y = 5 (a)

( 5,5)

(b)

( 2, −1)

(c)

176

(10,15)

(d)

(1, 2 )

2

4

x

Cumulative Review Test 1–4 Form B (cont.)

Name:

17. Solve the system: x − 2 z = −13 3 x + y = −7

5y − z = 5 (a) (3, –2, –5)

(b) (–3, 2, 5)

(c) (2, 5, –3)

(d) (–2, –5, 3)

18. Tricia worked 8 hours fewer this week than last. If she worked 32 hours this week, how many hours did she work last week? (a) 50 hr (b) 40 hr (c) 34 hr (d) 24 hr

Refer to the following problem for 19 and 20. Mr. Jarvis invests a total of $11,205 in two savings accounts. One account yields 9% simple interest and the other 8% simple interest. He would like to find the amount placed in each account if a total of $963.19 in interest is received after one year. 19. Which system of linear equations below expresses this problem? (a) x + y = 963.19 (b) x + y = 963.19 0.09 x + 0.08 y = 11, 205 9 x + 8 y = 11, 205 (c) x + y = 11, 205 0.09 x + 0.08 y = 963.19

(d) x + y = 11, 205 9 x + 8 y = 963.19

20. How much did Mr. Jarvis invest at 9%? (a) $362.08 (b) $601.11

(c) $4526

177

(d) $6679