algebra 2 3 linear systems and matrices practice problems ...rwright/algebra2/homework...algebra 2 3...
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Algebra 2 3 Linear Systems and Matrices Practice Problems
3.1 Solve Linear Systems by Graphing Graph the linear system and estimate the solution. Then check the solution algebraically.
1. {π¦ = β3π₯ + 2π¦ = 2π₯ β 3
2. {π¦ = βπ₯ + 3
βπ₯ β 3π¦ = β1
3. {π¦ = 2π₯ β 10π₯ β 4π¦ = 5
4. {π¦ = β3π₯ β 25π₯ + 2π¦ = β2
5. {π₯ β 7π¦ = 6
β3π₯ + 21π¦ = β18
6. {5π₯ β 4π¦ = 33π₯ + 2π¦ = 15
Graph and solve the system. Then classify the system as consistent and independent, consistent and dependent, or inconsistent.
7. {π¦ = β13π₯ + π¦ = 5
8. {π¦ = 3π₯ + 2π¦ = 3π₯ β 2
9. {β20π₯ + 12π¦ = β24
5π₯ β 3π¦ = 6
10. {3π₯ + 7π¦ = 62π₯ + 9π¦ = 4
11. {8π₯ + 9π¦ = 155π₯ β 2π¦ = 17
12. {3π₯ β 2π¦ = β15
π₯ β2
3π¦ = β5
Graph the system and estimate the solution(s). Then check the solution(s) algebraically.
13. {π¦ = |π₯ + 2|
π¦ = π₯
Problem Solving 14. You worked 14 hours last week and earned a total of $96 before taxes. Your job as a lifeguard pays $8 per
hour, and your job as a cashier pays $6 per hour. How many hours did you work at each job?
15. A gym offers two options for membership plans. Option A includes an initiation fee of $121 and costs $1 per
day. Option B has no initiation fee but costs $12 per day. After how many days will the total costs of the
gym membership plans be equal? How does your answer change if the daily cost of Option B increases?
Explain.
Mixed Review
16. (2-08) Graph π¦ > β1
3π₯ + 2
17. (2-04) Write the equation of the line that passes through (-2, 1) and (3, 5).
18. (1-07) Solve |2π₯ + 5| = 12
19. (1-04) Solve for y. 3π₯ β 2π¦ = 8
20. (1-03) Solve 8π₯ + 1 = 3π₯ β 14
Algebra 2 3 Linear Systems and Matrices Practice Problems
3.2 Solve Linear Systems Algebraically Solve the system using the substitution method.
1. {2π₯ + 5π¦ = 7
π₯ + 4π¦ = 2 2. {
3π₯ β π¦ = 26π₯ + 3π¦ = 14
3. {3π₯ + π¦ = β1
2π₯ + 3π¦ = 18
Solve the system using the elimination method.
4. {2π₯ + 6π¦ = 17
2π₯ β 10π¦ = 9 5. {
5π₯ β 3π¦ = β32π₯ + 6π¦ = 0
6. {3π₯ + 4π¦ = 186π₯ + 8π¦ = 18
7. Describe and correct the error in the first step of solving the system.
{3π₯ + 2π¦ = 75π₯ + 4π¦ = 15
Solve the system using any algebraic method.
8. {4π₯ β 10π¦ = 18β2π₯ + 5π¦ = β9
9. {3π₯ + π¦ = 15
βπ₯ + 2π¦ = β19
10. {2π₯ + π¦ = β1
β4π₯ + 6π¦ = 6
11. {
1
2π₯ +
2
3π¦ =
5
65
12π₯ +
7
12π¦ =
3
4
Use the elimination method to solve the system.
12. {7π¦ + 18π₯π¦ = 30
13π¦ β 18π₯π¦ = 90
Problem Solving 13. In one week, a music store sold 9 guitars for a total of $3611. Electric guitars sold for $479 each and
acoustic guitars sold for $339 each. How many of each type of guitar were sold?
14. An adult pass for a county fair costs $2 more than a childrenβs pass. When 378 adult and 214 childrenβs
passes were sold, the total revenue was $2384. Find the cost of an adult pass.
15. A nut wholesaler sells a mix of peanuts and cashews. The wholesaler charges $2.80 per pound for peanuts
and $5.30 per pound for cashews. The mix is to sell for $3.30 per pound. How many pounds of peanuts and
how many pounds of cashews should be used to make 100 pounds of the mix?
Mixed Review
16. (3-01) Solve by graphing: {3π₯ + π¦ = 11π₯ β 2π¦ = β8
17. (3-01) Solve by graphing: {π₯ β 2π¦ = β23π₯ + π¦ = β20
18. (2-02) Tell whether the lines are parallel, perpendicular, or neither:
Line 1: through (4, 5) and (9, -2)
Line 2: through (6, -6) and (-2, -1)
19. (1-07) Solve |π₯ + 3| = 4
20. (1-03) Solve 6(2π β 3) = β30
Algebra 2 3 Linear Systems and Matrices Practice Problems
3.3 Graph Systems of Linear Inequalities Graph the system of inequalities.
1. {βπ₯ + π¦ < β3βπ₯ + π¦ > 4
2. {4π₯ β 4π¦ β₯ β16βπ₯ + 2π¦ β₯ β4
3. {π¦ > |π₯| β 4
3π¦ < β2π₯ + 9
4. {2π¦ < β5π₯ β 105π₯ + 2π¦ > β2
5. {π₯ β 4π¦ β€ β10
π¦ β€ 3|π₯ β 1|
6. {π₯ < 6π¦ > β1π¦ < π₯
7. {
3π₯ + 2π¦ > β6β5π₯ + 2π¦ > β2
π¦ < 5
8. {π₯ β₯ 2
β3π₯ + π¦ < β14π₯ + 3π¦ < 12
9. {
π¦ β₯ 0π₯ > 3
π₯ + π¦ β₯ β2π¦ < 4π₯
10. {
π₯ β€ 10π₯ β₯ β2
3π₯ + 2π¦ < 66π₯ + 4π¦ > β12
Write a system of linear inequalities for the shaded region.
11.
Problem Solving 12. The Junior-Senior Banquet Committee must consist of 5 to 8 representatives from the junior and senior
classes. The committee must include at least 2 juniors and at least 2 seniors. Let x be the number of juniors
and y be the number of seniors.
a. Writing a System Write a system of inequalities to describe the situation.
b. Graphing a System Graph the system you wrote in part (a).
c. Finding Solutions Give two possible solutions for the numbers of juniors and seniors on the prom
committee.
Mixed Review
13. (3-02) Solve {9π₯ + 4π¦ = β73π₯ β 5π¦ = β34
14. (3-02) Solve {π₯ β 5π¦ = 18
2π₯ + 3π¦ = 10
15. (1-06) Solve π₯ β 8 β€ β5
Algebra 2 3 Linear Systems and Matrices Practice Problems
3.4 Solve Systems of Linear Equations in Three Variables 1. Write a linear equation in three variables. What is the graph of such an equation?
Tell whether the given ordered triple is a solution of the system. 2. (6, 0, -3)
{π₯ + 4π¦ β 2π§ = 123π₯ β π¦ + 4π§ = 6βπ₯ + 3π¦ + π§ = β9
Solve the system using the elimination method.
3. {
3π₯ + π¦ + π§ = 14βπ₯ + 2π¦ β 3π§ = β9
5π₯ β π¦ + 5π§ = 30 4. {
5π₯ + π¦ β π§ = 6π₯ + π¦ + π§ = 2
3π₯ + π¦ = 4
Solve the system using the substitution method.
5. {
π₯ + π¦ β π§ = 43π₯ + 2π¦ + 4π§ = 17βπ₯ + 5π¦ + π§ = 8
6. {2π₯ β π¦ + π§ = β2
6π₯ + 3π¦ β 4π§ = 8β3π₯ + 2π¦ + 3π§ = β6
Describe and correct the error in the first step of solving the system.
{
ππ + π β ππ = ππππ + ππ + π = ππ
π β π + π = βπ
7. Solve the system using any algebraic method.
8. {π₯ + 5π¦ β 2π§ = β1βπ₯ β 2π¦ + π§ = 6
β2π₯ β 7π¦ + 3π§ = 7 9. {
2π₯ β π¦ + 2π§ = β21π₯ + 5π¦ β π§ = 25
β3π₯ + 2π¦ + 4π§ = 6 10. {
π₯ + π¦ + π§ = 33π₯ β 4π¦ + 2π§ = β28βπ₯ + 5π¦ + π§ = 23
Problem Solving 11. The juice bar at a health club receives a delivery of juice at the beginning of each month. Over a three
month period, the health club received 1200 gallons of orange juice, 900 gallons of pineapple juice, and
1000 gallons of grapefruit juice. The table shows the composition of each juice delivery. How many gallons
of juice did the health club receive in each delivery?
Mixed Review
12. (3-03) Graph {
π¦ β₯ π₯π₯ β€ 4π¦ β₯ 1
13. (3-03) Graph {π¦ β€ β
1
2π₯ + 4
π¦ > π₯ β 3
14. (3-02) Solve the system using any algebraic method: {3π₯ β π¦ = β7
2π₯ + 3π¦ = 21
15. (2-02) Find the slope of the line passing through the given points. Then tell whether the line rises, falls, is
horizontal, or is vertical: (1, β4), (2, 6)
Algebra 2 3 Linear Systems and Matrices Practice Problems
3.5 Perform Basic Matrix Operations 1. Copy and complete: The _?_ of a matrix with 3 rows and 4 columns are 3Γ4.
Perform the indicated operation, if possible. If not possible, state the reason.
2. [10 β85 β3
] β [12 β33 β4
] 3. [
1.2 5.30.1 4.46.2 0.7
] + [2.4 β0.66.1 3.18.1 β1.9
] 4. [7 β3
12 5β4 11
] β [9 2
β2 66 5
]
Perform the indicated operation.
5. β3 [2 0 β54 7 β3
]
6. 1.5 [β2 3.4 1.65.4 0 β3
]
7. β2.2 [6 3.1 4.5
β1 0 2.55.5 β1.8 6.4
]
Use matrices A, B, C, and D to evaluate the matrix expression.
π¨ = [π βππ βπ
] π© = [ππ βππβπ π
] πͺ = [π. π βπ. π ππ. π
βπ. π π. π π] π« = [
π. π π βπ. ππ. π βπ. π π. π
]
8. π΅ β π΄
9. 2
3π΅
10. πΆ + 3π·
11. 0.5πΆ β π·
Solve the matrix equation for x and y.
12. [β2π₯ 6
1 β8] + 2 [
5 β1β7 6
] = [β9 4
β13 π¦] 13. 4π₯ [
β1 23 6
] = [8 β16
β24 3π¦]
14. Prove one of the properties of matrix operations on page 188 for 2Γ2 matrices. (Hint: Apply any related
properties of real numbers from page 3.)
Problem Solving 15. A sporting goods store sells snowboards in several different styles and lengths. The matrices below show
the number of each type of snowboard sold in 2003 and 2004. Write a matrix giving the change in sales for
each type of snowboard from 2003 to 2004.
Mixed Review
16. (3-04) Solve {
2π₯ β π¦ β 3π§ = 5π₯ + 2π¦ β 5π§ = β11
βπ₯ β 3π¦ = 10 17. (3-04) Solve {
2π₯ β 4π¦ + 3π§ = 16π₯ + 2π¦ + 10π§ = 19β2π₯ + 5π¦ β 2π§ = 2
18. (3-03) Graph {π¦ < 6π₯ + π¦ > β2
19. (2-08) Check whether the ordered pairs are solutions of the inequality: π₯ + 2π¦ β€ β3; (0, 3), (-5, 1)
20. (2-07)
Algebra 2 3 Linear Systems and Matrices Practice Problems
3.6 Multiply Matrices State whether the product AB is defined. If so, give the dimensions of AB.
1. π΄: 2 Γ 1, π΅: 2 Γ 2
2. If A is a 2Γ3 matrix and B is a 3Γ2 matrix, what are the dimensions of AB?
(A) 2Γ2 (B) 3Γ3 (C) 3Γ2 (D) 2Γ3
Find the product. If the product is not defined, state the reason.
3. [14
] [β2 1]
4. [9 β30 2
] [0 14 β2
]
5. [5 20 β41 6
] [3 7
β2 0] 6. [
1 3 02 12 β4
] [9 14 β3
β2 4]
Using the given matrices, evaluate the expression.
π¨ = [π βπ
βπ π] , π© = [
π ππ βπ
] , πͺ = [βπ ππ π
] , π« = [π π π
βπ π ππ π βπ
] , π¬ = [βπ π ππ π βππ π βπ
]
7. β1
2π΄πΆ
8. π΄π΅ β π΅π΄
9. (π· + πΈ)π·
10. 4π΄πΆ + 3π΄π΅
Problem Solving 11. Write an inventory matrix and a cost per item matrix. Then use matrix multiplication to write a total cost
matrix. A softball team needs to buy 12 bats, 45 balls, and 15 uniforms. Each bat costs $21, each ball costs
$4, and each uniform costs $30.
12. Matrix S gives the numbers of three types of cars sold in February by two car dealers, dealer A and dealer B.
Matrix P gives the profit for each type of car sold. Which matrix is defined, SP or PS? Find this matrix and
explain what its elements represent.
Mixed Review
13. (3-05) Simplify [3 β22 5
] + 2 [1 0
β4 0]
14. (3-02) Solve {3π₯ β 5π¦ = 112π₯ + 5π¦ = 24
15. (2-04) Write the equation of the line with slope: -3 and passes through (5, 2)
Algebra 2 3 Linear Systems and Matrices Practice Problems
3.7 Evaluate Determinants and Apply Cramerβs Rule Evaluate the determinant of the matrix.
1. [2 β14 β5
]
2. [β4 31 β7
]
3. [10 β6β7 5
]
4. [9 β37 2
]
5. [β1 12 40 2 β53 0 1
]
6. [5 0 2
β3 9 β21 β4 0
]
7. [12 5 80 6 β81 10 4
]
8. [β2 6 08 15 34 β1 7
]
Find the area of the triangle with the given vertices. 9. π΄(4, 2), π΅(4, 8), πΆ(8, 5) 10. π΄(β4, β4), π΅(β1, 2), πΆ(2, β6) 11. π΄(β6, 1), π΅(β2, β6), πΆ(0, 3)
Use Cramerβs rule to solve the linear system.
12. {3π₯ + 5π¦ = 3βπ₯ + 2π¦ = 10
13. {5π₯ + π¦ = β40
2π₯ β 5π¦ = 11
14. {
βπ₯ β 2π¦ + 4π§ = β28π₯ + π¦ + 2π§ = β11
2π₯ + π¦ β 3π§ = 30
15. {5π₯ β π¦ β 2π§ = β6π₯ + 3π¦ + 4π§ = 162π₯ β 4π¦ + π§ = β15
16. {
3π₯ β π¦ + π§ = 25βπ₯ + 2π¦ β 3π§ = β17
π₯ + π¦ + π§ = 21
Problem Solving 17. You are planning to turn a triangular region of your yard into a garden. The vertices of the triangle are
(0, 0), (5, 2), and (3, 6) where the coordinates are measured in feet. Find the area of the triangular region.
Mixed Review
18. (3-06) Simplify [2 β46 1
] [β3 01 7
]
19. (3-06) Simplify [1 03 β2
] [β5 102 0
]
20. (3-03) Graph {π₯ + π¦ β₯ 3
4π₯ + π¦ < 4
Algebra 2 3 Linear Systems and Matrices Practice Problems
3.8 Use Inverse Matrices to Solve Linear Systems Find the inverse of the matrix.
1. [1 β5
β1 4]
2. [6 25 2
]
3. [β4 β64 7
]
4. [β24 60β6 30
]
Solve the matrix equation.
5. [1 14 5
] π = [2 3
β1 6] 6. [
β1 06 4
] π = [3 β14 5
] 7. [1 50 β2
] π = [3 β1 06 8 4
]
Use an inverse matrix to solve the linear system.
8. {4π₯ β π¦ = 10
β7π₯ β 2π¦ = β25
9. {3π₯ β 2π¦ = 56π₯ β 5π¦ = 14
10. {β2π₯ β 9π¦ = β24π₯ + 16π¦ = 8
11. {6π₯ + π¦ = β2
βπ₯ + 3π¦ = β25
Problem Solving 12. A pilot has 200 hours of flight time in single-engine airplanes and twin-engine airplanes. Renting a single-
engine airplane costs $60 per hour, and renting a twin-engine airplane costs $240 per hour. The pilot has
spent $21,000 on airplane rentals. Use an inverse matrix to find how many hours the pilot has flown each
type of airplane.
Mixed Review 13. (3-07) You are making a triangular sail for a sailboat. The vertices of the sail are (0, 2), (12, 2), and (12, 26)
where the coordinates are measured in feet. Find the area of the sail.
14. (3-07) Evaluate the determinant of [5 4
β2 β3]
15. (3-06) Evaluate 2 [1 β45 2
] [2 β30 1
]
Algebra 2 3 Linear Systems and Matrices Practice Problems
3.Review Solve. Show some work.
1. {π₯ β 5π¦ = 10
π₯ = π¦ + 2
2. {10π₯ β 3π¦ = 15
β10π₯ + 5π¦ = 21
3. {π₯ + 7π¦ β 2π§ = 10
π¦ + 3π§ = 2π§ = 1
4. {π₯ β π¦ + π§ = 2
2π₯ + π¦ = 4βπ₯ + 2π¦ β 3π§ = β6
5. [3 π₯
4π¦ 2] = [
3 1512 2
]
Perform the indicated operation.
6. [35
] β [5
10]
7. 10 [2 97 β3
]
8. 3 [2 31 β1
] + 2 [0 β2
β1 3]
9. [1 23 4
] [β4 β3β2 β1
]
10. [2 4] [3 β2
β1 0]
Evaluate the determinate of the matrix.
11. [10 β120 2
] 12. [
1 3 52 4 60 β1 β2
]
Use Cramerβs Rule to solve the linear system.
13. {2π₯ β 3π¦ = 6
π₯ + π¦ = 2
14. Find the inverse of [2 β1
β3 4].
Use an inverse matrix to solve the linear system.
15. {2π₯ β π¦ = 8
β3π₯ + 4π¦ = 1
16. For a fundraiser, a student sold a total of 20 tickets for $122. If child tickets are $5 and adult tickets are $7,
how many of each type of ticket did the student sell?
Graph the linear system and tell how many solutions it has. If there is exactly one solution, estimate the solution and check it algebraically.
17. {π¦ = 2π₯ β 3
π¦ = β1
2π₯ + 2
18. {π₯ β 2π¦ = β1
β3π₯ + π¦ = β2
Graph the system of linear inequalities.
19. {π¦ < 3
π₯ + 2π¦ > β2
20. {π¦ β€
2
3π₯ + 4
π¦ β₯2
3π₯ β 1
Algebra 2 3 Linear Systems and Matrices Practice Problems
Answers
3.1 1. (1, -1) 2. (4, -1) 3. (5, 0) 4. (-2, 4) 5. Infinitely many solutions 6. (3, 3) 7. (2, -1); consistent and independent 8. No solution; inconsistent 9. Infinitely many solutions;
consistent and dependent 10. (2, 0); consistent and independent 11. (3, -1); consistent and independent
12. Infinitely many solutions; consistent and dependent
13. No solution 14. Lifeguard: 6h, cashier: 8h 15. 11 days: the number of days will
decrease 16. 17. 4π₯ β 5π¦ = β13
18. β81
2, 3
1
2
19. π¦ =3
2π₯ β 4
20. -3
3.2 1. (6, -1)
2. (4
3, 2)
3. (-3, 8)
4. (7,1
2)
5. (β1
2,
1
6)
6. No solution
7. Failed to multiply the constant by β2
8. Infinitely many solutions 9. (7, -6)
10. (β3
4,
1
2)
11. (-1, 2)
12. (β1
9, 6)
13. 5 acoustic, 4 electric
14. $4.75 15. 80 lbs. of peanuts, 20 lbs. of
cashews 16. (2, 5) 17. (-6, -2) 18. Neither 19. -7, 1 20. -1
3.3 1. No solution
2.
3. 4. No solution
5.
6.
7.
8.
9.
10.
11. {
π¦ β€ 3π¦ β₯ β2π₯ β€ 4
π₯ β₯ β3
12. {
π₯ β₯ 2π¦ β₯ 2
π₯ + π¦ β€ 8π₯ + π¦ β₯ 5
; ;
Sample: 3Jr., 4Sr.; 4Jr., 4Sr. 13. (-3, 5) 14. (8, -2) 15. π₯ β€ 3
Algebra 2 3 Linear Systems and Matrices Practice Problems
3.4 1. Sample: π₯ + π¦ + π§ = 4; plane 2. Yes 3. (1, 5, 6) 4. (π₯, 4 β 3π₯, 2π₯ β 2) or
(1
2π§ + 1, β
3
2π§ + 1, π§)
5. (3, 2, 1) 6. (0, 0, -2) 7. In the second equation, the
coefficient of y was not multiplied by 2.
8. (π¦ β 11, π¦, 3π¦ β 5) or
(1
3π§ β
28
3,1
3π§ +
5
3, π§)
9. (-4, 5, -4) 10. (2, 6, -5) 11. 1st: 300 gal, 2nd: 750 gal, 3rd: 2050
gal
12.
13. 14. (0, 7) 15. 10; rises
3.5 1. dimensions
2. [β2 β52 1
]
3. [3.6 4.76.2 7.5
14.3 β1.2]
4. [β2 β514 β1
β10 6]
5. [β6 0 15
β12 β21 9]
6. [β3 5.1 2.48.1 0 β4.5
]
7. [β13.2 β6.82 β9.9
2.2 0 β5.5β12.1 3.96 β14.08
]
8. [13 β8β9 1
]
9. [12 β8β4 0
]
10. [23.4 β1.5 β5.6β2.5 β2.3 9.9
]
11. [β6.3 β0.75 10.7β6.5 3.6 β3.3
]
12. π₯ =19
2, π¦ = 4
13. π₯ = β2, π¦ = β16 14. Possible answer (for commutative
property of addition): π΄ =
[π ππ π
] , π΅ = [π ππ β
] ; π΄ + π΅ =
[π ππ π
] + [π ππ β
] =
[π + π π + ππ + π π + β
] = [π + π π + ππ + π β + π
] =
[π ππ β
] + [π ππ π
] = π΅ + π΄
15. [0 5 1 β1
β7 β1 β5 β21 β1 4 10
]
16. (2, -4, 1)
17. (β1
2, 1, 2)
18. 19. Not a solution; solution
20. π¦ = β3
2|π₯ β 1| + 1
3.6 1. Not defined 2. A
3. [β2 1β8 4
]
4. [β12 15
8 β4]
5. [11 358 0
β9 7]
6. [21 β874 β50
]
7. [21 β6
β14 1]
8. [β10 7β8 10
]
9. [β2 4 05 15 8
β16 17 36]
10. [β204 81160 β38
]
11. π΅ππ‘π π΅ππππ
ππππππππ [124515
],
π΅ππ‘ π΅πππ πππππππ
πΆππ π‘ [ 21 4 30];
πΆππ π‘πΌπ‘ππ [ 882 ]
12. PS; [62,400 57,575], it shows the profit for all of the cars sold by each dealer.
13. [5 β2
β6 5]
14. (7, 2) 15. π¦ = β3π₯ + 17
3.7 1. -6 2. 25 3. 8 4. 39 5. -206 6. -34 7. 1160 8. -480 9. 12
10. 21 11. 25 12. (-4, 3) 13. (-7, -5) 14. (6, -3, -7) 15. (0, 4, 1) 16. (8, 6, 7) 17. 12 ft2
18. [β10 β28β17 7
]
19. [β5 10
β19 30]
20.
Algebra 2 3 Linear Systems and Matrices Practice Problems
3.8
1. [β4 β5β1 β1
]
2. [1 β1
β5
23 ]
3. [β
7
4β
3
2
1 1]
4. [β
1
12
1
6
β1
60
1
15
]
5. [11 9β9 β6
]
6. [β3 111
2β
1
4
]
7. [18 19 10β3 β4 β2
]
8. (3, 2) 9. (-1, -4) 10. (10, -2) 11. (1, -8) 12. Single: 150 h, twin: 50 h 13. 144 ft2 14. -7
15. [4 β14
20 β26]
3.Review 1. (0, β2)
2. (69
10, 18)
3. (19, β1, 1) 4. (1, 2, 3) 5. (15, 3)
6. [β2β5
]
7. [20 9070 β30
]
8. [6 51 3
]
9. [β8 β5
β20 β13]
10. [2 β4] 11. 40 12. 0
13. (12
5, β
2
5)
14. [
4
5
1
53
5
2
5
]
15. (33
5,
26
5)
16. 9 child, 11 adult 17. (2, 1);
18. (1, 1);
19.
20.