Deck 10: Systems of Equations

Question 1

The perimeter of a rectangle is 78 inches. The width of the rectangle is 9 inches less than the length of the rectangle. Find the dimensions of the rectangle.

A)The dimensions of the rectangle are 15 inches by 24 inches.
B)The dimensions of the rectangle are 14 inches by 23 inches.
C)The dimensions of the rectangle are 13 inches by 22 inches.
D)The dimensions of the rectangle are 16 inches by 25 inches.
E)The dimensions of the rectangle are 17 inches by 26 inches.

Answer 1

The dimensions of the rectangle are 15 inches by 24 inches.

Question 2

Use two equations in two variables to find the following integers. The sum of two integers is 34 and their difference is 8.

A)8, 26
B)11, 19
C)13, 19
D)13, 21
E)11, 26

Answer 2

13, 21

Question 3

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $$\left( \begin{array} { l } \frac { 1 } { 4 } x - \frac { 1 } { 20 } y = 10 \\\frac { 1 } { 4 } x + \frac { 1 } { 8 } y = - 11\end{array} \right)$$

A)(18, -122)
B)(16, -120)
C)(18, -120)
D)dependent
E)inconsistent

Answer 3

(16, -120)

Question 4

Use the substitution method to solve the following system. If the equations of the system are dependent, or if the system is inconsistent, so indicate. $$\left\{ \begin{array} { c } y = 4 x \\x + y = 5\end{array} \right.$$

A) $( 1,4 )$
B) $( 5,2 )$
C) $( 4,1 )$
D)The equations are dependent.
E)The system is inconsistent.

Question 5

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $$\left( \begin{array} { c } 4 x - y = 5 \\8 x + 6 y = 2\end{array} \right)$$

A)(1, -1)
B)(-1, 1)
C)(1, 1)
D)dependent
E)inconsistent

Question 6

Find the solution set of the system of inequalities. $$\left\{ \begin{array} { l } x + y < - 6 \\x - y > - 6\end{array} \right.$$

A)
B)
C)

Question 7

Use the addition method to solve the system. If the equations of the system are dependent, or if a system is inconsistent, so indicate. $$\left\{ \begin{array} { l } 6 ( x + 6 ) + 5 ( y - 4 ) = 6 \\6 ( x - 1 ) = - 5 ( y + 2 )\end{array} \right.$$

A) $( 4,5 )$
B) $( 1 , - 7 )$
C) $( - 4,1 )$
D)The equations are dependent.
E)The system is inconsistent.

Question 8

Solve the system. $$\left\{ \begin{array} { c } x + 4 y = - 4 \\x - 4 y = - 12\end{array} \right.$$

A) $( 1 , - 8 )$
B) $( 2,8 )$
C) $( - 8,1 )$
D) $( 8,1 )$
E) $( 2 , - 8 )$

Question 9

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $$\left( \begin{array} { c } 6 x - 2 y = 6 \\12 x - 4 y = - 5\end{array} \right)$$

A)(1, 2)
B)(3, -1)
C)(3, 2)
D)dependent
E)inconsistent

Question 10

Find the solution set of the system of inequalities. $$\left\{ \begin{array} { c } x - y \geq 6 \\x + 2 y < - 4\end{array} \right.$$

A)
B)
C)
D)
E)

Question 11

Use the substitution method to solve the following system. If the equations of the system are dependent, or if the system is inconsistent, so indicate. $$\left\{ \begin{array} { c } 7 a + 7 b = 42 \\a = 46 - 9 b\end{array} \right.$$

A) $( 1,5 )$
B) $( 1 , - 5 )$
C) $( - 1,5 )$
D)The equations are dependent.
E)The system is inconsistent.

Question 12

Solve the system. $$\left\{ \begin{array} { c } 3 x - 6 y = 18 \\x = 2 y + 3\end{array} \right.$$

A)
B)
C)
D)
E)

Question 13

Use two equations in two variables to find the following integers: One integer is twice another. Their sum is 39.

A)13, 26
B)26, 52
C)13, 52
D)16, 23
E)no such integers exist

Question 14

Use two equations in two variables to find the following integers. Twice one integer plus another integer is 19. The first integer plus 3 times the second is 32.

A)3, 7
B)5, 9
C)6, 9
D)6, 10
E)3, 10

Question 15

Use two equations in two variables to solve the following problem. At a theater, the giant rectangular movie screen has a width $24$ feet less than its length. If its perimeter is $308$ feet, find the area of the screen.

A) $5,655 \mathrm { ft } ^ { 2 }$
B) $5,740 \mathrm { ft } ^ { 2 }$
C) $5,720 \mathrm { ft } ^ { 2 }$
D) $5,893 \mathrm { ft } ^ { 2 }$
E) $5,785 \mathrm { ft } ^ { 2 }$

Question 16

Use the substitution method to solve the following system. If the equations of the system are dependent, or if the system is inconsistent, so indicate. $$\left\{ \begin{array} { l } 9 ( x - 1 ) + 41 = 75 - 7 y \\7 ( x + 1 ) + 29 = 68 - 4 y\end{array} \right.$$

A) $( 4,1 )$
B) $( 1,4 )$
C) $( - 4 , - 1 )$
D)The equations are dependent.
E)The system is inconsistent.

Question 17

Solve the system. $$\left\{ \begin{array} { l } x + y = 2 \\x - y = 0\end{array} \right.$$

A)
B)
C)
D)
E)

Question 18

Find the solution set of the system of inequalities. $$\left\{ \begin{array} { c } 5 x + 3 y > - 8 \\4 x - 5 y \geq 1\end{array} \right.$$

A)
B)
C)
D)
E)

Question 19

Use the addition method to solve the system. $$\left\{ \begin{array} { c } a + b = 3 \\a - b = - 1\end{array} \right.$$

A) $( 1 , - 2 )$
B) $( 2,1 )$
C) $( 1,2 )$
D) $( - 1,2 )$
E) $( - 2 , - 1 )$

Question 20

Use the addition method to solve the system. If the equations of the system are dependent, or if a system is inconsistent, so indicate. $$\left\{ \begin{array} { l } 2 ( x - 4 ) = 8 y \\4 ( 2 y + 2 ) = 2 x\end{array} \right.$$

A) $( 1 , - 3 )$
B) $( 2,2 )$
C) $( - 6,1 )$
D)The equations are dependent.
E)The system is inconsistent.

Question 21

Solve the system of equations by graphing. $$\left\{ \begin{array} { c } x ^ { 2 } + y ^ { 2 } = 98 \\x + y = 14\end{array} \right.$$

A) $( - 49,8 )$
B) $( 49 , - 49 )$
C) $( 8,5 )$
D) $( 5,8 )$
E) $( 7,7 )$

Question 22

Determine the graph of the system of equations. $$\left\{ \begin{array} { l } x ^ { 2 } + y ^ { 2 } = 2 \\- x + y = 2\end{array} \right.$$

A)
B)
C)
D)
E)

Question 23

Solve the system. Give your answer as an ordered triple in the form of ( a, b, c ). $$\left\{ \begin{array} { c } 8 a - 3 b = 1 \\6 a - 10 c = 1 \\2 b - 8 c = 0\end{array} \right.$$

A) $\left( - \frac { 1 } { 4 } , 1 , - \frac { 1 } { 4 } \right)$
B) $\left( \frac { 1 } { 4 } , - 1 , \frac { 1 } { 4 } \right)$
C) $\left( - \frac { 1 } { 4 } , - 1 , - \frac { 1 } { 4 } \right)$
D)The system is inconsistent.
E)The equations are dependent.

Question 24

Part of $7,000 is invested at 10%, another part at 11%, and the remainder at 12% yearly interest. The total yearly income from the three investments is $795. The sum of the amounts invested at 10% and 11% equals the amount invested at 12%. How much is invested at each rate?

A)$1,000 at 10%, $2,500 at 11%, and $3,500 at 12%
B)$500 at 10%, $2,000 at 11%, and $4,000 at 12%
C)$1,500 at 10%, $2,000 at 11%, and $3,500 at 12%
D)$500 at 10%, $2,500 at 11%, and $4,000 at 12%
E)$1,500 at 10%, $3,000 at 11%, and $3,000 at 12%

Question 25

Solve using substitution:

Question 26

Part of $9,000 is invested at 11%, another part at 13%, and the remainder at 15% yearly interest. The total yearly income from the three investments is $1,210. The sum of the amounts invested at 11% and 13% equals the amount invested at 15%. Determine how much is invested at each rate, if 2,500 is invested at first rate.

A)$2,500 at 11%, $2,000 at 13%, and $4,500 at 15%
B)$3,000 at 11%, $1,000 at 13%, and $4,000 at 15%
C)$3,500 at 11%, $1,000 at 13%, and $4,500 at 15%
D)$3,000 at 11%, $2,000 at 13%, and $4,000 at 15%
E)$2,000 at 11%, $3,000 at 13%, and $3,500 at 15%

Question 27

Determine the graph of the system of equations. $$\left\{ \begin{array} { c } x ^ { 2 } + y ^ { 2 } = 25 \\x - y = - 1\end{array} \right.$$

A)
B)
C)
D)

Question 28

Find the number of points of intersection of the system $$\left\{ \begin{array} { c } x ^ { 2 } + y ^ { 2 } = 49 \\x - y = 36\end{array} \right.$$

A)four
B)two
C)zero
D)one

Question 29

Solve the system. If a system is inconsistent or if the equations are dependent, so indicate.

Question 30

Part of $3,800 is invested at 12%, another part at 13%, and the remainder at 14%. The total yearly income from the three investments is $506. The sum of the amounts invested at 12% and 13% equals the amount invested at 14%. Determine how much is invested at each rate.

A)$1,100 at 12%, $800 at 13%, $1,900 at 14%
B)$700 at 12%, $1,200 at 13%, $1,900 at 14%
C)$700 at 12%, $1,900 at 13%, $1,200 at 14%
D)$600 at 12%, $1,300 at 13%, $1,900 at 14%
E)$800 at 12%, $1,300 at 13%, $1,700 at 14%

Question 31

Solve the system of equations by graphing. $$\left\{ \begin{array} { c } x ^ { 2 } + y ^ { 2 } = 25 \\52 x ^ { 2 } + 13 y ^ { 2 } = 676\end{array} \right.$$

A) $( 3,4 ) , ( - 3,4 )$
B) $( 2,4 ) , ( - 2,4 ) , ( 2 , - 4 ) , ( - 2 , - 6 )$
C) $( 3,4 ) , ( - 3,4 ) , ( 3 , - 4 ) , ( - 3 , - 4 )$
D) $( 2,6 ) , ( - 2,4 )$
E) $( 3,6 ) , ( - 3,6 ) , ( 3 , - 6 ) , ( - 3 , - 6 )$

Question 32

Solve the system of equations by graphing. $$\left\{ \begin{array} { c } x ^ { 2 } + y ^ { 2 } = 68 \\y = 2 x ^ { 2 }\end{array} \right.$$

A) $( 1,10 ) , ( - 2,10 )$
B) $( 2,8 ) , ( - 2,8 )$
C) $( - 2,8 )$
D) $( 2,8 )$
E) $( 2,10 ) , ( - 18 )$

Question 33

Use two equations in two variables to solve the following problem. Maria and Susan pool their resources to buy several lottery tickets. They win $400,000! They agree that Susan should get $50,000 more than Maria, because she gave most of the money. How much will Maria get?

A)$230,000
B)$225,000
C)$75,000
D)$175,000
E)$90,000