Deck 15: Systems of Equations: Matrices and Determinants

Question 1

The elimination-by-addition method and the graphing method can be used to obtain exact solutions for any system of two linear equations in two unknowns.

Answer 1

False

Question 2

Solve the problem by using a system of equations. The tens digit of a two-digit number is 7 less than three times the units digit. If the sum of the digits is 9, find the number.

A) 57
B) 54
C) 45
D) 48

Answer 2

54

Question 3

An inconsistent system has exactly one solution.

Answer 3

False

Question 4

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $\left( \begin{array} { l } x = 3 y + 3 \\x = - 4 y + 5\end{array} \right)$

A) $\left( \frac { 31 } { 7 } , \frac { 2 } { 7 } \right)$
B) $\left( \frac { 31 } { 7 } , - \frac { 2 } { 7 } \right)$
C) $\left( \frac { 27 } { 7 } , \frac { 2 } { 7 } \right)$
D) $\left( \frac { 29 } { 7 } , \frac { 2 } { 7 } \right)$
E) $\left( \frac { 27 } { 7 } , - \frac { 2 } { 7 } \right)$

Question 5

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $\left( \begin{array} { lll } y &=& 3 x \\0.31 x &+& 0.09 y &= &232\end{array} \right)$

A) (600, 1,800)
B) (400, 1,200)
C) dependent
D) inconsistent

Question 6

Michelle can enter a small business as a full partner and receive a salary of $40,000 a year and 15% of the year's profit, or she can be sales manager for a salary of $75,000 plus 5% of the year's profit. What must the year's profit be for her total earnings to be the same whether she is a full partner or a sales manager?

A) $\text { Year's profit } = \$ 320,000$
B) $\text { Year's profit } = \$ 310,000$
C) $\text { Year's profit } = \$ 340,000$
D) $\text { Year's profit } = \$ 350,000$
E) $\text { Year's profit } = \$ 330,000$

Question 7

Solve the problem by using a system of equations. The sum of the digits of a two-digit number is 17. If the digits are reversed, the newly formed number is 9 larger than the original number. Find the original number.

A) 90
B) 98
C) 99
D) 89

Question 8

Solve the problem by using a system of equations. The tens digit of a two-digit number is 17 less than three times the units digit. If the sum of the digits is 11, find the number.

Question 9

Solve the system by using the elimination-by-addition method. If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer as an ordered pair ( x , y ).

Question 10

Solve the problem by using a system of equations. The sum of the digits of a two-digit number is 7. If the digits are reversed, the newly formed number is 45 larger than the original number. Find the original number.

Question 11

Solve the system by using the substitution method. If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer as an ordered pair ( x , y ).

Question 12

A video store rents new release movies for $5 and favorites for $2.55. One day the number of new release movies rented was twice the number of favorites. If the total income for that day was $815.75, how many movies of each kind were rented?

A) 140 new release, 70 favorites
B) 150 new release, 75 favorites
C) 70 new release, 140 favorites
D) 65 new release, 130 favorites
E) 130 new release, 65 favorites

Question 13

Michelle can enter a small business as a full partner and receive a salary of $40,000 a year and 10% of the year's profit, or she can be sales manager for a salary of $75,000 plus 5% of the year's profit. What must the year's profit be for her total earnings to be the same whether she is a full partner or a sales manager? Year's profit = $__________

Question 14

Solve the system by using the elimination-by-addition method. $\left( \begin{array} { l } 3 x + 6 y = 3 \\5 x - 6 y = 1\end{array} \right)$

A) $\left( - \frac { 1 } { 4 } , - \frac { 1 } { 4 } \right)$
B) $\left( \frac { 1 } { 4 } , \frac { 1 } { 4 } \right)$
C) $\left( - \frac { 1 } { 2 } , \frac { 1 } { 4 } \right)$
D) $\left( \frac { 1 } { 2 } , - \frac { 1 } { 4 } \right)$
E) $\left( \frac { 1 } { 2 } , \frac { 1 } { 4 } \right)$

Question 15

One day last summer, Jim went kayaking on the Little Susitna River in Alaska. Paddling upstream against the current, he traveled 12 miles in 3 hours. Then he turned around and paddled twice as fast downstream and, with the help of the current, traveled 17 miles in 1 hour. Find the rate of the current. __________ miles per hour

Question 16

Solve the system by using substitution. If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer as an ordered pair ( x , y ).

Question 17

One day last summer, Jim went kayaking on the Little Susitna River in Alaska. Paddling upstream against the current, he traveled 24 miles in 3 hours. Then he turned around and paddled twice as fast downstream and, with the help of the current, traveled 25 miles in 1 hour. Find the rate of the current.

A) 14 mph
B) 8 mph
C) 11 mph
D) 3 mph

Question 18

When the graphs for a system of two linear equations are parallel lines, then there is no solution.

Question 19

Solve the system by using substitution. If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer as an ordered pair ( a , b ).

Question 20

A system of two linear equations can be solved by graphing the lines on the same set of axes.

Question 21

Solve the system. $\begin{array}{l}(4 x+3 y-3 z=29) \\x-5 z=12 \\5 x+5 z=0 \\\end{array}$

A) x = 2, y = 5, z = -2
B) x = 3, y = 5, z = -3
C) x = 3, y = 4, z = -2
D) x = 2, y = 6, z = -3

Question 22

A video store rents new release movies for $5 and favorites for $2.65. One day the number of new release movies rented was twice the number of favorites. If the total income for that day was $822.25, how many movies of each kind were rented? __________ new release __________ favorites

Question 23

Solve the system. $$\left( \begin{array} { r l } 3 x - 2 y + 3 z & = 23 \\3 y - 5 z & = - 18 \\2 z & = 6\end{array} \right)$$

A) x = 6, y = -2, z = 0
B) x = 5, y = 1, z = 0
C) x = 4, y = -1, z = 3
D) x = 6, y = -4, z = 3

Question 24

Solve the system. $\left(\begin{array}{rll}x+2 y-5 z & = & -20 \\3 y-z & = & -1 \\3 y+3 z & = & 15\end{array}\right)$

A) x = -2, y = 1, z = 4
B) x = -3, y = 3, z = 7
C) x = 1, y = 0, z = 7
D) x = -3, y = 4, z = 4

Question 25

Solve the system. $$\left( \begin{array} { r l r } - 2 x + 3 y + 5 z & = 23 \\3 x - 5 z & = - 13 \\5 z & = 25\end{array} \right)$$

A) x = 4, y = 2, z = 5
B) x = 5, y = 1, z = 3
C) x = 3, y = -1, z = 3
D) x = 3, y = 0, z = 5

Question 26

Solve the system. $\left(\begin{array}{rl}3 x+3 y-3 z & =-12 \\5 y+4 z & =17 \\3 y-5 z & =-12\end{array}\right)$

A) x = -4, y = 4, z = 2
B) x = -2, y = 0, z = 6
C) x = 1, y = 3, z = 2
D) x = -2, y = 1, z = 3

Question 27

A system of three linear equations in three variables produces three intersecting planes when graphed.

Question 28

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer as an ordered pair ( x , y ).

Question 29

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer as an ordered pair ( x , y ).

Question 30

In a system of three linear equations, if two of the planes coincide the solution is infinitely many solutions.

Question 31

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer as an ordered pair ( x , y ).

Question 32

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer as an ordered pair ( x , y ).

Question 33

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer as an ordered pair ( t , u ).

Question 34

Solve the system by using the elimination-by-addition method. If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer as an ordered pair ( x , y ).

Question 35

Solve the system by using the elimination-by-addition method. If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer as an ordered pair ( x , y ).

Question 36

To solve the system $$\left( \begin{array} { r } 2 x - y + 3 z = 5 \\2 y + z = 3 \\4 z = 8\end{array} \right)$$ , solve the equation $4 z = 8$ first.

Question 37

The solution set is infinitely many ordered triples if the three planes have a common line of intersection..

Question 38

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer as an ordered pair ( x , y ).

Question 39

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer as an ordered pair ( x , y ), enter x and y as fractions.

Question 40

Solve the system by using the elimination-by- addition method. If the equations of the system are dependent, or if a system is inconsistent, so indicate. In those cases enter dependent or inconsistent . Otherwise, enter your answer as an ordered pair ( x , y ).

Question 41

Solve the system. $$\left( \begin{array} { r l l } 5 x &+ &3 y& - &2 z & = & - 4 \\4 x& -& 7 y &+&7 z & = & 68 \\2 x &-& 5 y &+ &3 z & = & 36\end{array} \right)$$

A) x = 2, y = -5, z = 2
B) x = 3, y = -3, z = 5
C) x = 2, y = -6, z = 4
D) x = 4, y = -5, z = 4

Question 42

Solve the system.

Question 43

A gift store is making a mixture of almonds, pecans, and peanuts, which sell for $3.50 per pound, $5.00 per pound, and $1.00 per pound, respectively. The storekeeper wants to make 15 pounds of the mix to sell at $2.30 per pound. The number of pounds of peanuts is to be three times the number of pounds of pecans. Find the number of pounds of each to be used in the mixture.

A) 6 pounds of almonds, 6 pounds of pecans, and 8 pounds of peanuts
B) 9 pounds of almonds, 2 pounds of pecans, and 9 pounds of peanuts
C) 6 pounds of almonds, 6 pounds of pecans, and 12 pounds of peanuts
D) 3 pounds of almonds, 3 pounds of pecans, and 9 pounds of peanuts

Question 44

Solve the system. $$\left( \begin{array} { rrrrrr } 4 x &+ &3 y& -& z & = &- 8 \\&&3 x &-& 5 y & = &12 \\&&4 x &+& 5 y & = & - 19\end{array} \right)$$

A) x = 2, y = 0, z = -7
B) x = 2, y = -2, z = -7
C) x = -1, y = -3, z = -5
D) x = -1, y = -5, z = -2

Question 45

Solve the system.

Question 46

Solve the system.

Question 47

Solve the system.

Question 48

A small company makes three different types of bird houses. Each type requires the services of three different departments, as indicated by the following table. Type A Type B Type C Cutting department 0.2 hour 0.1 hour 0.3 hour Finishing department 0.5 hour 0.5 hour 0.4 hour Assembly department 0.1 hour 0.1 hour 0.3 hour The cutting, finishing, and assembly departments have available a maximum of 61, 128, and 52 work-hours per week, respectively. How many bird houses of each type should be made per week so that the company is operating at full capacity?

A) 70 houses of type A, 60 houses of type B, and 100 houses of type C
B) 90 houses of type A, 70 houses of type B, and 120 houses of type C
C) 80 houses of type A, 60 houses of type B, and 100 houses of type C
D) 90 houses of type A, 60 houses of type B, and 110 houses of type C

Question 49

Solve the system. $$\left( \begin{array} { c } 2 x& - &y &+& z& =& - 6 \\3 x& -& 2 y &+& 5 z& =& - 17 \\3 x &+ &y &-& 5 z& = &9\end{array} \right)$$

A) x = 4, y = 5, z = -1
B) x = 2, y = 4, z = -1
C) x = -1, y = 8, z = 0
D) x = -1, y = 2, z = -2

Question 50

Solve the system.

Question 51

Solve the system. $\left(\begin{array}{rlll}2 x&+&5 y&-&3 z & =&-13 \\5 x&-&2 y&+&4 z & =&34 \\&&4 y&+&z & =&-10\end{array}\right)$

A) x = 0, y = -5, z = 1
B) x = 4, y = -3, z = 2
C) x = 3, y = -6, z = 4
D) x = 0, y = -4, z = 4

Question 52

Solve the system. $\left(\begin{array}{llll}x&-&2 y&+&z & =&13 \\5 x&+&4 y&-&4 z & =&5 \\-2 x&-&5 y&+&4 z & =&13\end{array}\right)$

A) x = 5, y = -3, z = 2
B) x = 3, y = -1, z = 5
C) x = 5, y = 0, z = 4
D) x = 5, y = 2, z = 5

Question 53

Part of $10,000 is invested at 9%, another part at 11%, and the remainder at 12% yearly interest. The total yearly income from the three investments is $1,100. The sum of the amounts invested at 9% and 11% equals the amount invested at 12%. How much is invested at each rate?

A) $2,500 at 9%, $2,500 at 11%, and $5,000 at 12%
B) $1,500 at 9%, $2,500 at 11%, and $6,000 at 12%
C) $1,500 at 9%, $2,000 at 11%, and $6,000 at 12%
D) $3,000 at 9%, $2,000 at 11%, and $5,000 at 12%

Question 54

Solve the system. $$\left( \begin{array} { r l } x + 2 y - z & = 9 \\x + 3 y + 2 z & = 11 \\- x - 3 y - 2 z & = - 11\end{array} \right)$$

A) $( 7 k + 5 , - 3 k + 2 , k )$ , $k$ is a real number
B) $( 6 k + 6 , - 2 k + 2 , k )$ , $k$ is a real number
C) $( 7 k + 5 , - 2 k + 2 , k )$ , $k$ is a real number
D) $( 6 k + 6 , - 3 k + 2 , k )$ , $k$ is a real number
E) $( 6 k + 5 , - 3 k + 2 , k )$ , $k$ is a real number

Question 55

Solve the system. $\left(\begin{array}{rlll}4 x&+&2 y&-&z & = & -4 \\4 x&-&4 y&+&4 z & =&40 \\5 x&+&y&-&5 z & = & -14\end{array}\right)$

A) x = 5, y = -2, z = 2
B) x = 7, y = -4, z = 2
C) x = 2, y = -1, z = 6
D) x = 2, y = -4, z = 4

Question 56

Solve the system. $$\left( \begin{array} { r l l } 2 x&-&4 y&+&2 z&=&0 \\4 x&+&2 y&-&6 z&=&20 \\5 x&-&3 y&+&6 &z=&-10\end{array} \right)$$

A) x = 3, y = -2, z = -6
B) x = -1, y = 1, z = -6
C) x = 1, y = -4, z = -4
D) x = 1, y = -1, z = -3

Question 57

A box contains $6.80 in nickels, dimes, and quarters. There are 40 coins in all, and the sum of the numbers of nickels and dimes is two less than the number of quarters. How many coins of each kind are there?

A) 7 nickels, 12 dimes, and 21 quarters
B) 11 nickels, 11 dimes, and 18 quarters
C) 6 nickels, 13 dimes, and 21 quarters
D) 6 nickels, 10 dimes, and 25 quarters

Question 58

The measure of the largest angle of a triangle is twice the measure of the smallest angle. The sum of the smallest angle and the largest angle is twice the other angle. Find the measure of each angle.

A) 55^o, 60^o, 65^o
B) 45^o, 60^o, 75^o
C) 30^o, 60^o, 90^o
D) 40^o, 60^o, 80^o

Question 59

Solve the system. $$\left( \begin{array} { r l } 3 x - 5 y - 2 z & = 24 \\x - y + z & = 7 \\- 3 x + 4 y - 2 z & = - 24\end{array} \right)$$

A) x = -1, y = -6, z = 2
B) x = -1, y = -7, z = 2
C) x = -4, y = -3, z = -1
D) x = 2, y = -4, z = 1

Question 60

Solve the system. $\left(\begin{array}{llll}x&-&5 y&+&2 z & = & 24 \\2 x&+&2 y&+&4 z & = & 12 \\2 x&-&2 y&-&3 z & = & -4\end{array}\right)$

A) x = 1, y = -6, z = 3
B) x = 0, y = -4, z = 1
C) x = 0, y = -7, z = 1
D) x = 1, y = -3, z = 4

Question 61

The measure of the largest angle of a triangle is twice the measure of the smallest angle. The sum of the smallest angle and the largest angle is twice the other angle. Find the measure of each angle. Please enter your answer as an ordered triple ( x , y , z ), where x , y , z are the measures (in degrees) of the smallest, medium, and the largest angle, respectively. Please do not include the units in your answer.

Question 62

Solve the system.

Question 63

Part of $5,000 is invested at 9%, another part at 10%, and the remainder at 11% yearly interest. The total yearly income from the three investments is $505. The sum of the amounts invested at 9% and 10% equals the amount invested at 11%. How much is invested at each rate? Please enter your answer as an ordered triple ( x , y , z ), where x , y , z are the amounts invested at 9%, 10%, and 11%, respectively. Please do not include units in your answer. $__________, $__________, $__________

Question 64

Solve the system.

Question 65

Solve the system.

Question 66

Solve the system.

Question 67

The matrix is in reduced echelon form. $$\left. \left[ \begin{array} { l l } 1 & 0 \\0 & 1\end{array} \right] \quad \begin{array} { l l } 13 \\3\end{array} \right]$$

Question 68

The augmented matrix of a system of equations is a matrix of the coefficients of the equations.

Question 69

Solve the system.

Question 70

The matrix $\left( \begin{array} { l } 3 x + y - z = 6 \\2 y + 3 z = 4 \\4 z = 10\end{array} \right)$ is in echelon form.

Question 71

A gift store is making a mixture of almonds, pecans, and peanuts, which sell for $2.50 per pound, $4.00 per pound, and $1.00 per pound, respectively. The storekeeper wants to make 20 pounds of the mix to sell at $1.90 per pound. The number of pounds of peanuts is to be three times the number of pounds of pecans. Find the number of pounds of each to be used in the mixture. Please enter your answer as an ordered triple ( x , y , z ), where x , y , z are the numbers of pounds of almonds, pecans, and peanuts, respectively.

Question 72

Transformations that are applied to augemented matrices are called elemtary row operations.

Question 73

Solve the system.

Question 74

A box contains $7.40 in nickels, dimes, and quarters. There are 44 coins in all, and the sum of the numbers of nickels and dimes is two less than the number of quarters. How many coins of each kind are there? Please enter your answer as an ordered triple ( x , y , z ), where x , y , z are the numbers of nickels, dimes, and quarters, respectively.

Question 75

A matrix of dimension $3 \times 4$ has 3 rows and 4 columns.

Question 76

Solve the system.

Question 77

Solve the system.

Question 78

Solve the system.

Question 79

Solve the system.

Question 80

A small company makes three different types of bird houses. Each type requires the services of three different departments, as indicated by the following table. Type A Type B Type C Cutting department 0.1 hour 0.1 hour 0.2 hour Finishing department 0.5 hour 0.5 hour 0.3 hour Assembly department 0.3 hour 0.1 hour 0.2 hour The cutting, finishing, and assembly departments have available a maximum of 27, 86, and 37 work-hours per week, respectively. How many bird houses of each type should be made per week so that the company is operating at full capacity? Please enter your answer as an ordered triple ( x , y , z ), where x , y , z are the numbers of houses of type A, B, and C, respectively.