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Skills Practice Skills Practice for Lesson 2.1
Name _____________________________________________ Date ____________________
Finding a Job Introduction to Systems of Linear Equations
Vocabulary Write the term that best completes each statement.
1. A(n) is the location on a graph where two or more
lines meet indicating that the values are the same.
2. A(n) is two or more linear equations in the same
variables.
Problem Set Complete each table. Then determine the point where the two lines intersect.
3. 4.
x y � 2 x y � 3x x y � x � 3 y � 2 x
�1 1
0 2
1 3
2 4
5. 6.
x y � �x y � x � 6 x y � x � 4 y � �x
0 0
3 1
6 2
9 3
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7. 8.
x y � 2 __ 5 x y � �x � 7 x y � 1 __
2 x � 2 y � x � 4
�10 �6
�5 �4
0 �2
5 0
9. 10.
x y � �x � 2 y � x � 6 x y � 2 x � 4 y � 4x � 2
�3 �2
�2 �1
�1 0
0 1
Graph both equations on the same coordinate grid. Determine the point where the two lines intersect.
11. y � 3x and y � 5x 12. y � �2x and y � 4x
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13. y � x � 1 and y � �x � 3 14. y � 2 x � 5 and y � �x � 4
15. y � 4x � 2 and y � �6x � 8 16. y � x � 5 and y � 2x � 7
For each situation, explain what the point of intersection of the two equations represents.
17. The cost in dollars of making x DVDs is given by the equation y � 2x � 7000. The
revenue r in dollars of selling x DVDs is given by y � 12x.
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18. A police car needs to catch a speeding car. The speeding car is already 10 miles ahead of
the police car. The distance the speeding car travels is given by the equation y � 100x �
10, where x represents the time in hours. The distance the police car travels is given by the
equation y � 110x, where x represents the time in hours.
19. The amount of water a plant needs in liters after t months is given by the equation
y � 0.1t � 2. The amount of water that a planter can hold in liters after t months is
given by the equation y � 4.
20. The capacity of a garage after t months is given by the equation y � 750. The
number of cars in the garage after t months is given by the equation y � 15t � 50.
21. The amount of hard drive space in gigabytes that a computer uses t months after
it was bought is given by the equation y � 20t � 30. The amount of space left on
the hard drive in gigabytes t months after it was bought is given by the equation
y � �20t � 220.
22. The cost to buy one gigabyte of hard drive space in dollars after t months is given
by the equation y � �0.25t � 4. The cost to make one gigabyte of hard drive
space in dollars after t months is given by the equation y � �0.10t � 3.
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Skills Practice Skills Practice for Lesson 2.2
Name _____________________________________________ Date ____________________
Pens-R-Us Solving Systems of Linear Equations: Graphing and Substitution
Vocabulary Give an example of how to solve a system of two equations by using the given method.
1. substitution
2. graphing
Problem Set Use the slope and y-intercept to graph the equations. Then determine the point of intersection.
3. � y � x � 3
y � �x � 1 4. � y � x � 5
y � �2x � 4
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5. � y � x � 2
y � 3x � 4 6. � y � 4x � 4
y � �x � 9
7.
� y � 1 __ 2
x � 5
y � x � 7
8. � y � 1 __
5 x � 4
y � �x � 2
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9.
� y � 2 __ 3 x � 6
y � 2x � 2
10. � y � 2 __
3 x � 3
y � �3x � 8
11. � y � 6x � 5
y � �3x � 13 12. � y � 1.5x � 8
y � �3x � 10
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Use algebraic substitution to solve each system of equations.
13. � �x � y � 2
x � y � 4
14. � �2x � y � �1
�x � y � 2
15. � �x � y � �1
2x � y � �4 16. � �2x � y � 4
x � y � �5
17. � x � �y � 10
y � x � 2
18. � x � y � 8
y � �2x � 7
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19. � y � 3x � 1
�2x � y � 3 20. � y � x � 7
x � y � 8
21. � y � 1.5x � 170
y � 2x � 120
22. � y � 2.3x � 565
y � 3.8x � 1015
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Skills Practice Skills Practice for Lesson 2.3
Name _____________________________________________ Date ____________________
Tickets Solving Systems of Linear Equations: Linear Combinations
Vocabulary For the two equations x � 3y � 7 and 4x � y � 2, explain how you would use linear combinations to eliminate each variable.
1. Eliminate the x variable.
2. Eliminate the y variable.
Problem Set Use the slope and y-intercept to graph the equations. Then determine the point of intersection.
3. � y � �1 __ 3 x � 5
y � �2 __
3 x � 3
4. � y � 1 __ 2
x � 1
y � �1 __ 4 x � 2
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5. � y � 1 __ 6 x � 1
y � � 1 __
3 x � 2
6. � y � 1 __ 4
x � 3 __ 2
y � 3 __ 2
x � 4
Use algebraic substitution to solve each system of equations.
7. � 2 x � y � 9
x � 2y � 12 8. � x � 3y � 7
2x � 4y � 12
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9. � 3x � 2y � 3
2x � 4y � 10 10. � 4x � 2y � �4
10x � 3y � 8
Use linear combinations to solve each system of equations.
11. � 2x � y � �5
x � 2y � 2
12. � x � y � 6
2x � y � 8
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13. � 2 x � 2y � 5
3x � y � 5
14. � �3x � 2y � �12
x � 2y � �4
15. � 3x � 4y � �13
2x � 3y � �9 16. � 2 x � 3y � �14
5x � 4y � �7
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17. � 45x � 120y � 15
4x � 6y � 20
18. � �230x � 50y � 1140
7x � 2y � �28
19. �
1 __ 5 x � 1 __
4 y � �2
� 3 __
5 x � 1 __
2 y � 3
20. �
2 __ 3 x � 2 __
3 y � �16
1 __ 6 x � 1 ___
10 y � �3
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Chapter 2 ● Skills Practice 61
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Skills Practice Skills Practice for Lesson 2.4
Name _____________________________________________ Date ____________________
Cramer’s Rule Solving Systems of Linear Equations: Cramer’s Rule
Vocabulary Write the term that best completes each statement.
1. Graphing, substitution, linear combinations, and Cramer’s Rule are all methods for
solving .
2. An algebraic operation that transforms an n � n array of numbers into a single
value is called calculating the .
3. uses determinants to solve a system of linear
equations in two variables.
4. A(n) array is made up of n � n numbers.
Problem Set Calculate the value of each determinant.
5. | 4 1 5 2
|
6. | 3 9 1 6
|
7. | �2 4 3 8
|
8. | �10 �3 �1 2
|
9. | 12 2 6 1
|
10. | �8 6 �4 3
|
11. | �2 1 __
3
1 __ 4 1 __
3 |
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62 Chapter 2 ● Skills Practice
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12. |
3 ___ 10
1 __ 2
� 1 __ 5 �2
| 13. | 0 �0.2
�0.7 3 |
14. | 0.68 144 0 0.27
|
User Cramer’s Rule to solve each system of equations.
15. 2x � 3y � 2
�x � 2y � 2
16. 3x � 4y � 12
x � 2y � �26
17. 0.125x � 0.2y � 4
x � 2y � �4
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18. 0.2x � 3y � 12
�0.3x � 2y � 1
19. 3y � 27
2x � 1 __ 3 y � �1
20. 1 __ 4 x � 2y � 10
2x � 16
21. y � 5x � 1
y � �5x � 3
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22. y � �4x � 1
y � 8x � 4
23. 2x � 3y � 4
�x � 9y � 3
24. 2 x � 8y � 4
�3x � 16y � �3
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Skills Practice Skills Practice for Lesson 2.5
Name _____________________________________________ Date ____________________
Consistent and Independent Solving Systems of Linear Equations: Consistent and Independent
Vocabulary Define each term in your own words.
1. consistent system of equations
2. inconsistent system of equations
3. linearly independent system of equations
4. linearly dependent system of equations
Problem Set Choose any method to solve each system of equations.
5. � y � x � 4
y � �2 x � 5 6. � y � x � 5
y � 2 x � 2
7. � �x � y � �2
x � y � 6
8. � �x � y � �7
x � y � 5
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9. � �3x � 2y � �12
x � 2y � �4
10. � �2 x � 4y � �4
�x � 4y � 4
11. � 2 x � 3y � 13
�2y � �10 12. � �3x � �9
2 x � y � �3
13. � � 1 __ 3 x � 3 __
2 y � 17
� 2 __
3 x � 1 __
2 y � �1
14. � 1 __ 4 x � 2 __
5 y � 1
�3 __
4 x � 4 __
5 y � �1
Solve each system of equations. Then describe the system’s consistency and dependency.
15. � �x � 2y � 6
x � 2y � 2
16. � �2x � y � 3
2x � y � 2
17. � x � 2y � 1
�2x � 4y � �2
18. � 3x � y � 2
6x � 2y � 4
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19. � 3x � 4y � �6
�6x � 8y � �26
20. � 4x � 6y � �2
2x � 9y � �61
21. � y � 2x � 1
4x � 2y � �2
22. � y � �3x � 2
6x � 2y � 4
23. � y � �x � 2
2x � 2y � �8
24. � y � �x � 2
12x � 3y � �15
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Skills Practice Skills Practice for Lesson 2.6
Name _____________________________________________ Date ____________________
Inequalities—Infinite Solutions Solving Linear Inequalities and Systems of Linear Inequalities in Two Variables
Vocabulary Give an example of each term.
1. linear inequality in two variables
2. equation of a half-plane
3. system of two linear inequalities in two variables
4. inequality
Problem Set Graph each inequality.
5. y � x � 4 6. y � �x � 2
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7. y � 2x � 3 8. y �3x � 1
9. y � 1 __ 2 x � 1 10. y � � 2 __
3 x � 2
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Graph the solution to each system of linear inequalities.
11. � y � 2x � 1
y � x � 3 12. � y � 3x � 2
y 5x � 2
13. � y � �3x � 2
y � x � 2
14. � y �x � 1
y �4x � 5
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15. � y � 2x � 10
y � �3x � 15 16. � y � �x � 4
y � �3x � 12
17. � y � 1 __ 2 x � 8
y 2 __
3 x � 9
18. � y � 3 __ 4
x � 5
y � 1 __ 8 x � 5
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19. � y 1 __ 5 x � 1
y � 1 __
2 x � 4
20. � y � 1 __ 3
x � 6
y � 1 __ 9 x � 4
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Skills Practice Skills Practice for Lesson 2.7
Name _____________________________________________ Date ____________________
Three in Three or More Solving Systems of Three or More Linear Equations in Three or More Variables
Vocabulary Explain how the two terms are related.
1. 3 � 3 determinant and a 3 � 3 square array
Problem Set Evaluate each 3 � 3 determinant.
2. 1 3 4
1 1 2
2 1 1
3. 2 1 1
1 5 1
1 1 8
4. 3 2 1
0 1 1
4 0 5
5. 4 0 1
2 6 3
1 0 5
6. �1 0 4
2 2 �3
3 0 1
7. 0 �5 2
2 �1 3
3 0 0
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8.
1 __ 2 0 3
�1 1 4
0 3 __ 2 0
9.
1 __ 4 �2 0
0 3 1
1 __ 5 2 __
3 0
10.
1 __ 3 2 �0
4 1 2 __ 3
�2 � 1 __ 3 1
11.
1 __ 4 2 8
�4 � 1 __ 4 4
3 __ 4
�2 0
Write a system of three linear equations in three variables to model each situation. Define each variable.
12. The sum of the ages of Marise, Sophia, and Jaren is 45 years. Jaren is six years
older than Marise. Sophia is three years older than Marise.
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13. The sum of three basketball players' heights, Amna, Wen, and Nireta, is 186 inches.
Nireta is 2 inches taller than Wen. Nireta is 7 inches taller than Amna.
14. An office supply store sells flash drives for $20 each, headphones for $15 each,
and reams of paper for $25 each. Roger spent a total of $130 on flash drives,
headphones, and reams of paper. It cost him $115 for the flash drives and reams of
paper. It cost him $90 for the headphones and reams of paper.
15. At the movies a bag of popcorn costs $3, a box of chocolate-covered raisins costs
$2.50, and a bottle of water costs $2. Erika spent $14 on popcorn, raisins, and
bottles of water for her and her friends. The popcorn and raisins totaled $8. The
raisins and bottles of water totaled $11.
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Use substitution to solve each system of equations.
16.
x � 2y � z � 2
�3x � y � z � �2
2x � 3y � z � 5
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17.
2 x � y � z � 4
x � 2y � z � 1
x � 2y � z � �1
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18.
x � 2y � 2 z � �8
2x � y � z � �1
�x � 3y � 2 z � 13
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19.
2x � y � z � �3
�x � 3y � 2z � 4
x � 2y � 2z � �1
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Use linear combinations to solve each system of equations.
20.
2y � z � 1
2 x � 3y � 5z � �1
�x � y � 3z � �2
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21.
3x � y � 9
�x � 2y � 3z � 8
5x � 4y � 3z � �10
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22.
� 3 __ 4 x � 2 __
3 y � 2 __
3 z � 3
x � 3y � 2z � �17
1 __ 2 x � y � z � 11
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23.
3 __ 4 x � 4 __
3 y � 1 __
5 z � 15
2x � 5y � 4z � �1
x � y � z � �1
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Use Cramer’s Rule to solve each system of equations.
24.
1.5x � 3.2y � 1.8z � 1.3
�x � 5y � 2z � �8
y � 2z � �1
25.
�0.3x � 2.7y � 0.5z � �0.5
2x � 3y � z � 8
2x � y � 10
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26.
2x � 3y � 5z � 20
�x � 10y � 5z � �3
3x � 2y � z � 9
27.
�x � 4y � 2z � 1
2x � 3y � z � �4
x � 2y � 3z � 5
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