Deck 4: Solving Systems of Equations and Inequalities

Question 1

Without graphing, determine the number of solutions of: $$\left\{ \begin{array} { l } 4 y = - 2 x + 4 \\4 x + 8 y = 5\end{array} \right.$$

A)No solutions
B)One solution
C)Two solutions
D)Infinitely many solutions

Answer 1

No solutions

Question 2

Refer to the illustration. Answer the questions. How many solutions does the system of equations have? Are the equations dependent or independent?

A)no solution, independent equations
B)no solution, dependent equations
C)infinite number of solutions, dependent equations
D)two solutions, independent equations
E)one solution, independent equations

Answer 2

no solution, independent equations

Question 3

Use the substitution method to solve the following system: $$\left\{ \begin{array} { c } 0.07 x + 0.04 y = 0.3 \\x = \frac { y } { 2 }\end{array} \right.$$

A)(-1, -2)
B)(0, 0)
C)(2, 4)
D)The equations are dependent.
E)The system is inconsistent.

Answer 3

(2, 4)

Question 4

Determine whether the ordered pair is a solution of the system of equations shown below. Ordered pair $( 2,4 )$ System of equations $x - y = - 2$ $3 x + 2 y = 14$

A)No
B)Yes
C)Cannot be determined

Question 5

Use the substitution method to solve the following system: $$\left\{ \begin{array} { l } 2 x = \frac { 1 } { 2 } y - 1.2 \\\frac { 1 } { 5 } y = 6 x - 1.6\end{array} \right.$$

A)(-1.6, 4)
B)(0.4, 4)
C)(0.5, 3)
D)The equations are dependent.
E)The system is inconsistent.

Question 6

Solve the system by graphing. $$\left\{ \begin{array} { c } x = 2 - 2 y \\2 x + 4 y = 4\end{array} \right.$$

A)
B)
C)
D)
E)

Question 7

Solve the system by graphing. $$\left\{ \begin{array} { l } 4 x = 4 - 3 y \\4 x + 3 y = 7\end{array} \right.$$

A)
B)
C)
D)
E)

Question 8

Consider the following system: $$\left\{ \begin{array} { c } 7 x + y = 54 \\y = - 7 x - 18\end{array} \right.$$ Substitute $- 7 x - 18$ for y in the first equation. How many variables does the resulting equation contain?

A)1
B)0
C)2
D)4
E)none of the above

Question 9

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 10

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. $$\left\{ \begin{array} { c } y = 5 x \\x + y = 6\end{array} \right.$$

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

Question 11

The ordered pair (1, 3)is a solution of the given system. $$\left\{ \begin{array} { l } 2 x + 2 y = 8 \\4 x + 3 y = 13\end{array} \right.$$

Question 12

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. $$\left\{ \begin{array} { l } 5 ( x - 1 ) + 23 = 43 - 3 y \\9 ( x + 1 ) + 35 = 82 - 4 y\end{array} \right.$$

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

Question 13

Solve the system by substitution, if possible. $$\left\{ \begin{array} { c } x + 2 y = 7 \\3 x - 2 y = - 14\end{array} \right.$$

A) $( - 3,5 )$
B) $( 5,3 )$
C) $( 3,5 )$
D)infinitely many solutions, dependent equations
E)no solution, inconsistent system

Question 14

Tell whether the ordered pair (- 5, 3)is a solution of the given system. $$\left\{ \begin{array} { l } - 3 x + 8 y = 36 \\4 x - 5 y = - 31\end{array} \right.$$

A)no
B)yes

Question 15

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. $$\left\{ \begin{array} { c } 7 a + 9 b = 69 \\a = 21 - 5 b\end{array} \right.$$

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

Question 16

Solve the system by any method, if possible. $$\left\{ \begin{array} { c } 0.18 x - 0.09 y = 0.36 \\2 x - 4 = y\end{array} \right.$$

A) $\left( \frac { 3 } { 2 } , 0.36 \right)$
B) $\left( 0.18 , \frac { 3 } { 2 } \right)$
C) $\left( 0.36 , \frac { 3 } { 2 } \right)$
D)infinitely many solutions, dependent equations
E)no solution, inconsistent system

Question 17

Without graphing, determine the number of solutions of: $$\left\{ \begin{array} { l } 2 x - 4 y = 6 \\x + 2 y = - 3\end{array} \right.$$

A)No solutions
B)One solution
C)Two solutions
D)Infinitely many solutions

Question 18

Without graphing, determine the number of solutions of: $$\left\{ \begin{array} { l } 6 x - y = - 3 \\18 x + 9 = 3 y\end{array} \right.$$

A)No solutions
B)One solution
C)Two solutions
D)Infinitely many solutions

Question 19

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 20

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

A)
B)
C)
D)
E)

Question 21

Use the addition method to solve the system. $$\left\{ \begin{array} { c } a + b = 5 \\a - b = - 1\end{array} \right.$$

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

Question 22

Use the addition method to solve the system. If the equations of the system are dependent, or if a system is inconsistent, so indicate. $$\left\{ \begin{array} { c } 3 ( x - 4 ) = 8 y \\4 ( 2 y + 3 ) = 3 x\end{array} \right.$$

A) $( 4 , - 7 )$
B) $( 3,3 )$
C) $( - 1,7 )$
D)The equations are dependent.
E)The system is inconsistent.

Question 23

Use two equations in two variables to solve the following problem. A rancher raises five times as many cows as horses. If he has 144 animals (cows and horses only), how many cows does he have?

A)120 cows
B)100 cows
C)160 cows
D)110 cows
E)90 cows

Question 24

Solve the system by any method, if possible. If the system is inconsistent, or if the equations are dependent, so indicate.

Question 25

Use the method of elimination to find the y -coordinate of the solution of the following system of equations. $$\left\{ \begin{array} { l } 4 x - 5 y = - 32 \\2 x + 5 y = 14\end{array} \right.$$

A) $y = - 4$
B) $y = 4$
C) $y = 3$
D) $y = - 3$

Question 26

Solve the system by elimination if possible. $$\left\{ \begin{array} { c } 2 x + 3 y = 29 \\3 x - 2 y = - 15\end{array} \right.$$

A)(1, 9)
B)(1, $- 9$ )
C)(9, 1)
D)infinitely many solutions, dependent equations
E)no solution, inconsistent system

Question 27

A tree casts a shadow of 26 feet at the same time as a 6-foot man casts a shadow of 4 feet. Find the height of the tree.

A)41 feet
B)35 feet
C)39 feet
D)31 feet
E)52 feet

Question 28

In solving a system of equations by the elimination method, the result 1 = 1 is obtained. Which of the following statements is true?

A)The system is consistent.
B)The system is inconsistent.
C)The equations are dependent.
D)There is only one solution.

Question 29

Solve the system by any method, if possible. If the system is inconsistent, or if the equations are dependent, so indicate.

Question 30

Use two equations in two variables to solve the following problem. Maria and Susan pool their resources to buy several lottery tickets. They win $700,000! They agree that Susan should get $50,000 more than Maria, because she gave most of the money. How much will Maria get?

A)$280,000
B)$275,000
C)$225,000
D)$325,000
E)$240,000

Question 31

Solve the system by any method, if possible. $$\left\{ \begin{array} { l } x = \frac { 3 } { 2 } y + 3 \\2 x - 3 y = 9\end{array} \right.$$

A) $\left( \frac { 1 } { 2 } , 12 \right)$
B) $\left( 12 , \frac { 3 } { 2 } \right)$
C) $\left( 9 , \frac { 2 } { 3 } \right)$
D)infinitely many solutions, dependent equations
E)no solution, inconsistent system

Question 32

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $$\left( \begin{array} { l } 8 x - y = - 25 \\9 x + 6 y = - 21\end{array} \right)$$

A)(-3, 1)
B)(0, -1)
C)(-3, -1)
D)dependent
E)inconsistent

Question 33

A box contains $7.85 in nickels, dimes, and quarters. There are 48 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)17 nickels, 8 dimes, and 23 quarters
B)16 nickels, 7 dimes, and 25 quarters
C)16 nickels, 12 dimes, and 28 quarters
D)14 nickels, 9 dimes, and 25 quarters
E)17 nickels, 11 dimes, and 20 quarters

Question 34

Use the addition method to solve the system. If the equations of the system are dependent, or if a system is inconsistent, so indicate. $$\left\{ \begin{array} { l } 2 ( x + 2 ) + 9 ( y - 4 ) = 1 \\2 ( x - 1 ) = - 9 ( y + 2 )\end{array} \right.$$

A) $( 8,4 )$
B) $( 7 , - 2 )$
C) $( - 3,5 )$
D)The equations are dependent.
E)The system is inconsistent.

Question 35

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $$\left( \begin{array} { c } 3 x - 8 y = 3 \\6 x - 16 y = - 2\end{array} \right)$$

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

Question 36

Solve the system by using either the substitution method or the elimination-by-addition method, whichever seems more appropriate. $$\left( \begin{array} { c } \frac { 1 } { 2 } x - \frac { 1 } { 6 } y = - 12 \\\frac { 1 } { 2 } x + \frac { 1 } { 4 } y = 8\end{array} \right)$$

A)(-10, 46)
B)(-8, 48)
C)(-10, 48)
D)dependent
E)inconsistent

Question 37

Use the dimensions in the illustration to find w , the width of the river. The two triangles in the illustration are similar.

A) $w = 52 \frac { 2 } { 3 }$ ft
B) $w = 54 \frac { 3 } { 7 }$ ft
C) $w = 51 \frac { 2 } { 3 }$ ft
D) $w = 54 \frac { 1 } { 7 }$ ft
E) $w = 51 \frac { 3 } { 7 }$ ft

Question 38

If the addition method is used to solve the following system, by what would we multiply the first equation to eliminate m ? $$\left\{ \begin{aligned}6 m + 3 y & = 9 \\48 m - 10 & = 18\end{aligned} \right.$$

A)2
B)8
C)-2
D)-8

Question 39

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) $5,669 \mathrm { ft } ^ { 2 }$
B) $5,764 \mathrm { ft } ^ { 2 }$
C) $5,738 \mathrm { ft } ^ { 2 }$
D) $5,873 \mathrm { ft } ^ { 2 }$
E) $5,785 \mathrm { ft } ^ { 2 }$

Question 40

Use two equations in two variables to solve the following problem. In a year the number of people dying from cancer was 6 times the number who died from accidents. If the number of deaths from these two causes totaled 280,000, how many people died from each cause?

A)220,000 died from cancer, 40,000 died from accidents
B)250,000 died from cancer, 30,000 died from accidents
C)220,000 died from cancer, 60,000 died from accidents
D)240,000 died from cancer, 40,000 died from accidents
E)250,000 died from cancer, 60,000 died from accidents

Question 41

Graph the solution of the system: $$\left\{ \begin{array} { l } y < - 4 x - 1 \\y > - 4 x + 2\end{array} \right.$$

A)
B)
C)
D)

Question 42

An airplane can fly with the wind a distance of 920 miles in 4 hours. However, the return trip against the wind takes 5 hours. Find the speed of the plane in still air and the speed of the wind.

A)211 mph, 19 mph
B)203 mph, 27 mph
C)207 mph, 22 mph
D)207 mph, 23 mph
E)230 mph, 23 mph

Question 43

Use two equations in two variables to solve the following problem. Two pairs of shoes and four pairs of socks cost $107, and three pairs of shoes and five pairs of socks cost $156. Find the cost of a pair of socks.

A)$6.50
B)$5.50
C)$6.00
D)$4.50
E)$4.00

Question 44

A merchant wishes to mix peanuts that sell for $2.5 per pound and cashews that sell for $8 per pound to get 100 pounds of mixed nuts that sell for $4.50 per pound. Let x equal the number of pounds of peanuts and y equal the number of pounds of cashews. If solving this by writing two equations in two variables, which of the following could be one of the equations?

A) $2.5 x + 8 y = 450$
B) $2.5 x + 8 y = 4.50$
C) $x + y = 4.5$
D) $2.5 x + 8 y = 100$

Question 45

Graph the solution of the system: $$\left\{ \begin{array} { l } y < 5 x - 3 \\y - 5 x \geq 4\end{array} \right.$$

A)
B)
C)
D)

Question 46

A boat traveled 48 miles downstream in 4 hours and made the return trip in 6 hours. Find the rate of the current.

A)10 mph
B)2 mph
C)3 mph
D)4 mph

Question 47

Part of $2,200 is invested at 12%, another part at 13%, and the remainder at 14%. The total yearly income from the three investments is $295. 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)$800 at 12%, $300 at 13%, $1,100 at 14%
B)$200 at 12%, $900 at 13%, $1,100 at 14%
C)$200 at 12%, $1,100 at 13%, $900 at 14%
D)$100 at 12%, $1,000 at 13%, $1,100 at 14%
E)$300 at 12%, $1,000 at 13%, $900 at 14%

Question 48

Melodic Music has compact discs on sale for either $15 or $20. If a customer wants to spend at least $60 but no more than $120 on CDs, graph a system of inequalities that will show the possible ways a customer can buy $15 CDs ( x )and $20 CDs ( y ).

A)
B)
C)
D)
E)

Question 49

Dry Boat Works wholesales aluminum boats for $500 and fiberglass boats for $300. Northland Marina wants to order at least $750 worth but no more than $1,500 worth of boats. Write a system of linear inequalities which describes all possible combinations of aluminum boats ( x )and fiberglass boats ( y )that can be ordered, graph the system of inequalities, and determine whether the ordered pairs satisfy the inequality.

A) $\left\{ \begin{array} { c } 500 x + 300 y < 1,500 \\500 x + 300 y > 750\end{array} \right.$ $( 1,2 ) , ( 10,5 )$
B) $\left\{ \begin{array} { c } 500 x + 300 y \leq 1,500 \\500 x + 300 y \leq 750\end{array} \right.$ $( 10,5 ) , ( 3,0 )$
C) $\left\{ \begin{array} { c } 500 x + 300 y \leq 1,500 \\500 x + 300 y \geq 750\end{array} \right.$ $( 1,2 ) , ( 3,0 )$
D) $\left\{ \begin{array} { c } 500 x + 300 y > 1,500 \\500 x + 300 y < 750\end{array} \right.$ $( 5,0 ) , ( 1,2 )$
E) $\left\{ \begin{array} { c } 500 x + 300 y \geq 1,500 \\500 x + 300 y \leq 750\end{array} \right.$ $( 6,9 ) , ( 5,0 )$

Question 50

In a mix of pennies and nickels, there are 22 coins with a total value of 82 cents. How many nickels are there?

A)11 nickels
B)7 nickels
C)8 nickels
D)15 nickels

Question 51

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 52

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 53

Doris invested some money at 7% and some money at 8%. She invested $4,000 more at 8% than she did at 7%. Her total yearly interest from the two investments was $920. How much did Doris invest at each rate?

A)$100 at 7%, $7,900 at 8%
B)$5,000 at 7%, $8,000 at 8%
C)$4,000 at 7%, $9,000 at 8%
D)$4,000 at 7%, $8,000 at 8%
E)$4,500 at 7%, $7,500 at 8%

Question 54

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 55

Graph the solution of the system: $$\left\{ \begin{array} { c } x - 2 y < 4 \\y < \frac { x } { 2 } - 5\end{array} \right.$$

A)
B)
C)
D)

Question 56

Use two equations in two variables to solve the following problem. The salaries of the president and vice president of the United States total $490,000 a year. If the President makes $210,000 more than the Vice president, find each of their salaries.

A)President: $280,000, Vice president: $112,000
B)President: $350,000, Vice president: $140,000
C)President: $420,000, Vice president: $168,000
D)President: $315,000, Vice president: $126,000

Question 57

A distributor wholesales desk chairs for $200 and side chairs for $100. Best Furniture wants to order no more than $1,200 worth of chairs and wants to order more side chairs than desk chairs. Find a system of inequalities whose graph will show the possible combination of desk chairs ( x )and side chairs ( y )that can be ordered, graph the system of inequalities, and determine whether the ordered pairs satisfy the inequality.

A) $\left\{ \begin{array} { c } 200 x + 100 y < 1,200 \\y \geq x\end{array} \right.$ $( 0,10 ) , ( 4,3 )$
B) $\left\{ \begin{array} { c } 200 x + 100 y \leq 1,200 \\y > x\end{array} \right.$ $( 0,10 ) , ( 0,11 )$
C) $\left\{ \begin{array} { c } 200 x + 100 y \leq 1,200 \\y < x\end{array} \right.$ $( 3,3 ) , ( 4,0 )$
D) $\left\{ \begin{array} { c } 200 x + 100 y \leq 1,200 \\y > x\end{array} \right.$ $( 0,3 ) , ( 11,0 )$
E) $\left\{ \begin{array} { c } 200 x + 100 y \geq 1,200 \\y > x\end{array} \right.$ $( 4,3 ) , ( 0,10 )$

Question 58

One day last summer, Jim went kayaking on the Little Susitna River in Alaska. Paddling upstream against the current, he traveled 45 miles in 5 hours. Then he turned around and paddled twice as fast downstream and, with the help of the current, traveled 27 miles in 1 hour. Find the rate of the current.

A)3 mph
B)27 mph
C)12 mph
D)15 mph
E)9 mph

Question 59

Part of $10,000 is invested at 9%, another part at 10%, and the remainder at 12% yearly interest. The total yearly income from the three investments is $1,075. The sum of the amounts invested at 9% and 10% equals the amount invested at 12%. How much is invested at each rate?

A)$2,500 at 9%, $2,500 at 10%, and $5,000 at 12%
B)$3,500 at 9%, $3,500 at 10%, and $4,000 at 12%
C)$1,500 at 9%, $3,500 at 10%, and $5,000 at 12%
D)$3,500 at 9%, $2,500 at 10%, and $4,000 at 12%
E)$1,500 at 9%, $1,500 at 10%, and $6,000 at 12%

Question 60

A gift store is making a mixture of almonds, pecans, and peanuts, which sell for $3.00 per pound, $4.00 per pound, and $2.00 per pound, respectively. The storekeeper wants to make 20 pounds of the mix to sell at $2.70 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. # of almonds = __________ # of pecans = __________ # of peanuts = __________