Deck 42: Two Variable Linear Systems

Question 1

Solve the system of linear equations by the method of elimination.Use the graph to check your solution.​ $$\left\{ \begin{array} { l } 2 x + y = 2 \\x - y = 4\end{array} \right.$$ ​​

A)(-2,2)
B)​ (-2,6) ​
C)(2,-2)​​
D)​(2,-6)​
E)​(-2,-6)

Answer 1

(2,-2)​​

Question 2

Solve the system of linear equations by the method of elimination.Use the graph to check your solution. ​​ $$\left\{ \begin{array} { c } 5 x - y = - 5 \\10 x + 3 y = - 10\end{array} \right.$$ ​

A)(-1,0)​ ​
​
B)​(0,1) ​
​
C)(10,1) ​
​
D)​(1,10)​
E)​(1,0) ​
​

Answer 2

(-1,0)​ ​
​

Question 3

Solve the system of linear equations by the method of elimination.Use the graph to check your solution.​ $$\left\{ \begin{array} { l } x + 4 y = 5 \\- x + 5 y = 4\end{array} \right.$$ ​

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

Answer 3

​(1,1)

Question 4

Solve the system of linear equations by the method of elimination.Find (x,y)and check your solution algebraically.​ $$\left\{ \begin{array} { l } x + 2 y = 6 \\x - 2 y = 2\end{array} \right.$$ ​

A)​(1,4)
B)​(4,1)
C)(-​4,-1)
D)(​4,-1)
E)​(1,-4)

Question 5

Find the equilibrium point (x,p)of the demand and supply equations.The equilibrium point is the price p and number of units x that satisfy both the demand and supply equations. ​
Demand
Supply $p = 150 - 0.07 x$ $p = 65 + 0.1 x$ ​

A)​(-500,-115)
B)​​(500,115)
C)​​(-500,115)
D)​(115,-500)
E)​​(115,500)

Question 6

​Use any method to solve the system of linear equations,find (x,y).​ $$\left\{ \begin{array} { l } x - 3 y = 25 \\4 x + 3 y = 25\end{array} \right.$$ ​

A)​(-10,-5)
B)(10,-5)
C)​(3,-5)
D)​​(-5,10)
E)​​​(-5,-10)

Question 7

Use any method to solve the system of linear equations,find (x,y).​ $$\left\{ \begin{array} { c } 4 x - 5 y = 13 \\5 x + y = 9\end{array} \right.$$ ​

A)​(-1,-2)
B)(-1,2)
C)​(-2,-1)
D)​(2,5)
E)​(2,-1)

Question 8

​Use any method to solve the system of linear equations,find (x,y).​ $$\left\{ \begin{aligned}- x + 3 y & = 15 \\2 x + 3 y & = 6\end{aligned} \right.$$ ​ ​

A)​(3,4)
B)​(4,-3)
C)​​(-3,-4)
D)​​(-3,4)
E)​​(4,-3)

Question 9

​Solve the system of linear equations by the method of elimination.Find (x,y)and check your solution algebraically. ​​ $$\left\{ \begin{array} { l } 9 x - 6 y = 6 \\6 x + 6 y = 24\end{array} \right.$$ ​

A)​(2,-2)
B)​(2,2)
C)​(2,9)
D)​(-2,-2)
E)​(-2,2)

Question 10

Use any method to solve the system of linear equations,find (x,y).​ $$\left\{ \begin{array} { l } y = 4 x - 7 \\y = 7 x - 13\end{array} \right.$$ ​

A)​​​(1,-2)
B)​​(1,2)
C)​(7,2)
D)​(-2,1)
E)(2,1)

Question 11

Find the equilibrium point (x,p)of the demand and supply equations.The equilibrium point is the price p and number of units x that satisfy both the demand and supply equations. ​
Demand
Supply $p = 410 - 0.0004 x$ $p = 230 + 0.0002 x$ ​

A)​(290,300,000)
B)​(290,-300,000) ​
C)(-300,000,-290) ​
D)(300,000,290)
E)(-300,000,290) ​

Question 12

Use any method to solve the system of linear equations,find (x,y). ​​ $$\left\{ \begin{array} { l } y = - 7 x - 12 \\y = 7 - 8 x\end{array} \right.$$ ​

A)​(145,19)
B)​​(-19,145)
C)​(19,-145)
D)​​(145,-19)
E)​​(-145,19)

Question 13

Find the equilibrium point (x,p)of the demand and supply equations.The equilibrium point is the price p and number of units x that satisfy both the demand and supply equations. ​
Demand
Supply $p = 570 - 0.5 x$ $p = 370 + 0.3 x$ ​

A)(250,445)​
B)​​(445,-250)​
C)​(445,250)​
D)​(-250,445)​
E)​(-250,-445)​

Question 14

Two cheeseburgers and one small order of French fries from a fast-food restaurant contain a total of 850 calories.Three cheeseburgers and two small orders of French fries contain a total of 1380 calories.Find the caloric content of each item. ​

A)Cheeseburger: 320 calories;French fries: 210 calories
B)​Cheeseburger: 210 calories;French fries: 320 calories
C)​Cheeseburger: 850 calories;French fries: 320 calories
D)​Cheeseburger: 210 calories;French fries: 210 calories
E)​Cheeseburger: 320 calories;French fries: 320 calories

Question 15

Solve the system of linear equations by the method of elimination.Use the graph to check your solution. ​​ $$\left\{ \begin{array} { c } x + y = 0 \\9 x + 8 y = 1\end{array} \right.$$ ​​

A)​(1,-1)​
B)​(0,1)​ ​
C)​(-1,1)​ ​
D)​(1,1)​ ​
E)​(-1,-1)​ ​

Question 16

One eight-ounce glass of apple juice and one eight-ounce glass of orange juice contain a total of 176.4 milligrams of vitamin C.Twoeight-ounce glasses of apple juice and three eight-ounce glasses of orange juice contain a total of 432.7 milligrams of vitamin C.How much vitamin C is in an eight-ounce glass of each type of juice? ​

A)​Apple juice: 96.5 mg;Orange juice: 96.5 mg
B)Apple juice: 96.5 mg;Orange juice: 79.9 mg
C)​Apple juice: 79.9 mg;Orange juice: 79.9 mg
D)​Apple juice: 79.9 mg;Orange juice: 96.5 mg
E)​Apple juice: 176.4 mg;Orange juice: 79.9 mg

Question 17

Find the equilibrium point (x,p)of the demand and supply equations.The equilibrium point is the price p and number of units x that satisfy both the demand and supply equations. ​
Demand
Supply $p = 140 - 0.00003 x$ $p = 90 + 0.00001 x$ ​

A)(103,1,250,000)
B)​(-1,250,000,-103)
C)​(1,250,000,103)
D)​(103,-1,250,000)
E)​(-1,250,000,103)

Question 18

Fourty liters of a 40% acid solution is obtained by mixing a 28% solution with a 50% solution.Write a system of equations in which one equation represents the amount of final mixture required and the other represents the percent of acid in the final mixture.Let x and y represent the amounts of the 28% and 50% solutions,respectively. ​

A)​ $$\left\{ \begin{array} { l } x + y = 40 \\0.28 x + 0.5 y = 40\end{array} \right.$$
B)​ $$\left\{ \begin{array} { l } 0.28 x + 0.5 y = 40 \\x + y = 16\end{array} \right.$$
C) $$\left\{ \begin{array} { l } x + y = 40 \\0.28 x + 0.5 y = 16\end{array} \right.$$

Question 19

Solve the system of linear equations by the method of elimination.Use the graph to check your solution.​ $$\left\{ \begin{array} { c } x - y = 2 \\- 2 x + 2 y = 9\end{array} \right.$$ ​

A)​(2,2)​ ​
B)​(2,9)​ ​
C)​(-9,9)​ ​
D)​(9,- 2)​ ​
E)No solution​ ​

Question 20

Solve the system of linear equations by the method of elimination.Use the graph to check your solution.​ $$\left\{ \begin{array} { l } 9 x + 4 y = 4 \\18 x + 8 y = 16\end{array} \right.$$ ​​

A)(9,2)​ ​
B)​(- 4,2) ​ ​
​
C)​(2,- 9) ​
D)​(- 2,- 4) ​
E)No solution ​

Question 21

Solve the system of linear equations by the method of elimination.Find (x,y)and check your solution algebraically.​ $$\left\{ \begin{array} { l } 0.2 x - 0.5 y = - 27.1 \\0.3 x + 0.4 y = 70.9\end{array} \right.$$ ​

A)(97,107)
B)(107,97)
C)(107,-97) ​
D)(-97,107) ​
E)(-107,97) ​

Question 22

Solve the system of linear equations by the method of elimination.Find (x,y)and check your solution algebraically.​ $$\left\{ \begin{array} { l } 4 b + 4 m = \frac { 156 } { 35 } \\\\4 b + 11 m = \frac { 471 } { 35 }\end{array} \right.$$ ​

A)​ $$\left( \frac { 6 } { 35 } , \frac { 9 } { 7 } \right)$$
B) $$\left( \frac { - 6 } { 35 } , \frac { 9 } { 7 } \right)$$
C)​ $$\left( \frac { 9 } { 7 } , \frac { 6 } { 35 } \right)$$
D)​ $$\left( \frac { 9 } { 7 } , \frac { - 6 } { 35 } \right)$$
E)​ $$\left( \frac { - 9 } { 7 } , \frac { - 6 } { 35 } \right)$$

Question 23

Seven hundred gallons of 87-octane gasoline is obtained by mixing 87-octane gasoline with 90-octane gasoline.Use a graphing utility to graph the two given equations.Let x and y represent the number of gallons of the 87-octane gasoline and 90-octane gasoline.As the number of gallons of the 87-octane gasoline increases,how does the number of gallons of the 90-octane gasoline change?​ $$\left\{ \begin{array} { l } x + y = 700 \\87 x + 90 y = 60,900\end{array} \right.$$ ​

A)​ Increases
B)​ Decreases
C)​ Decreases
D)​ Increases
E)​ Decreases

Question 24

Solve the system of linear equations by the method of elimination.Find (x,y)and check your solution algebraically. ​​ $$\left\{ \begin{array} { c } 5 x + 7 y = 21 \\- 14 x - 18 y = - 38\end{array} \right.$$ ​

A)​(14,13)
B)(-14,-13) ​
C)(-14,14)
D)(-14,13)
E)(14,-13)

Question 25

Use any method to solve the system of linear equations,find (x,y). ​​ $$\left\{ \begin{array} { c } 4 x - 3 y = 16 \\- 5 x + 3 y = - 23\end{array} \right.$$ ​

A)(7,4)
B)​(4,7)
C)​​(4,-7)
D)​(-7,4)
E)​(-7,-4)

Question 26

Solve the system of linear equations by the method of elimination.Find (x,y)and check your solution algebraically.​ $$\left\{ \begin{array} { c } 5 x + 3 y = 6 \\3 x - y = 5\end{array} \right.$$ ​

A)​ $\left( \frac { 3 } { 2 } , \frac { 1 } { 2 } \right)$
B)​ $$\left( - \frac { 3 } { 2 } , \frac { 1 } { 2 } \right)$$
C)​ $$\left( \frac { 3 } { 2 } , - \frac { 9 } { 2 } \right)$$
D)​ $\left( - \frac { 3 } { 2 } , - \frac { 1 } { 2 } \right)$
E) $\left( \frac { 3 } { 2 } , - \frac { 1 } { 2 } \right)$

Question 27

​Solve the system of linear equations by the method of elimination,find (x,y).​ $$\left\{ \begin{array} { c } - 5 x + 2 y = - 38 \\3 x - y = 23\end{array} \right.$$ ​ ​

A)No solution
B)​ $( 8,1 )$
C)​ $( 8 , - 1 )$
D)​ $\left( \frac { 3 } { 38 } , \frac { 2 } { 23 } \right)$
E)​ $\left( - 5 , \frac { 2 } { 23 } \right)$

Question 28

Solve the system of linear equations by the method of elimination.Find (x,y)and check your solution algebraically.​ $$\left\{ \begin{array} { r } x + 5 y = 11 \\3 x - 10 y = 18\end{array} \right.$$ ​

A) $\left( - 8 , \frac { 3 } { 5 } \right)$ ​
B) $\left( - 8 , - \frac { 3 } { 5 } \right)$ ​
C) $\left( 8 , \frac { 3 } { 5 } \right)$ ​
D) $\left( 8 , - \frac { 3 } { 5 } \right)$ ​
E) $\left( 8 , \frac { 5 } { 3 } \right)$ ​

Question 29

Solve the system of linear equations by the method of elimination.Find (x,y)and check your solution algebraically.​ $$\left\{ \begin{array} { l } 3 x + 11 y = 12 \\- 2 x - 5 y = 6\end{array} \right.$$ ​

A)​(18,6)
B)(-18,-6) ​
C)(-18,18)
D)(-18,6)
E)​(18,-6) ​

Question 30

Sixty liters of a 43% acid solution is obtained by mixing a 28% solution with a 42% solution.Use a graphing utility to graph the two given equations.Let x and y represent the amounts of the 28 % solution and 42% solution.As the amount of the 28% solution increases,how does the amount of the 42% solution change? $$\left\{ \begin{array} { l } x + y = 60 \\0.28 x + 0.42 y = 25.8\end{array} \right.$$ ​ ​
​

A)​ ​
Increases
B)​ ​
Decreases
C)​ ​
Decreases
D)​ ​
Decreases
E)​
Increases

Question 31

Solve the system by the method of elimination and check any solutions algebraically. ​​ $$\left\{ \begin{array} { l } 0.05 x - 0.03 y = 0.29 \\0.07 x + 0.02 y = 0.22\end{array} \right.$$ ​

A)(4,- 3)
B)(4,3)
C)​(7,- 7)
D)​(- 4,3)
E)​(- 4,- 3)

Question 32

Solve the system of linear equations by the method of elimination.Find (r,s)and check your solution algebraically.​ $$\left\{ \begin{array} { c } 2 r + 4 s = 7 \\16 r + 50 s = 77\end{array} \right.$$ ​

A)​ $\left( \frac { 7 } { 6 } , - \frac { 7 } { 6 } \right)$
B)​ $( 7,77 )$
C)​ $\left( \frac { 7 } { 6 } , \frac { 7 } { 6 } \right)$
D)​ $\left( - \frac { 7 } { 6 } , \frac { 7 } { 6 } \right)$
E)​ $\left( - \frac { 7 } { 6 } , - \frac { 7 } { 6 } \right)$

Question 33

A total of $23,000 is invested in two corporate bonds that pay 3.5% and 5% simple interest.The investor wants an annual interest income of $880 from the investments.What amount should be invested in the 3.5% bond? ​

A)$22,800
B)$18,000
C)$23,000
D)$18,300
E)$18,100

Question 34

A total of $32,000 is invested in two municipal bonds that pay 5.75% and 6.25% simple interest.The investor wants an annual interest income of $1,900 from the investments.What amount should be invested in the 5.75% bond? ​

A)$31,800
B)$20,000
C)$32,000
D)$20,300
E)$20,100

Question 35

Solve the system of linear equations by the method of elimination.Find (x,y)and check your solution algebraically.​ $$\left\{ \begin{array} { l } 2 x + 7 y = \frac { 112 } { 9 } \\\\7 x + 8 y = \frac { 62 } { 9 }\end{array} \right.$$ ​

A) $\left( \frac { - 14 } { 9 } , \frac { 20 } { 9 } \right)$
B)​ $\left( \frac { 20 } { 9 } , \frac { - 14 } { 9 } \right)$
C)​ $\left( \frac { 20 } { 9 } , \frac { 14 } { 9 } \right)$
D)​ $\left( \frac { 14 } { 9 } , \frac { 20 } { 9 } \right)$
E)​ $\left( \frac { 14 } { 9 } , \frac { - 20 } { 9 } \right)$

Question 36

Thirty​ liters of a 40% acid solution is obtained by mixing a 28% solution with a 43% solution.How much of each solution is required to obtain the specified concentration of the final mixture? Use this system of linear equations there x and y represents the amounts of the 28% solution and 43% solution. ​​ $$\left\{ \begin{array} { l } x + y = 30 \\0.28 x + 0.43 y = 12\end{array} \right.$$ ​

A)​28% solution: 6 L;43% solution: 24 L
B)43% solution: 6 L;28% solution: 24 L​
C)​28% solution: 40 L;43% solution: 24 L
D)​43% solution: 40 L;28% solution: 24 L
E)​43% solution: 7 L;28% solution: 24 L

Question 37

​Seven hundred gallons of 89-octane gasoline is obtained by mixing 87-octane gasoline with 92-octane gasoline.How much of each type of gasoline is required to obtain the 700 gallons of 89-octane gasoline? Use this system of linear equations there x and y represent the number of gallons of the 87-octane gasoline and 89-octane gasoline. ​​ $$\left\{ \begin{array} { c } x + y = 700 \\87 x + 92 y = 62300\end{array} \right.$$ ​

A)87​-octane: 89 gal;92-octane: 280 gal ​
B)87​-octane: 280 gal;92-octane: 420 gal​
C)92​​-octane: 89 gal;87-octane: 280 gal
D)​​87-octane: 420 gal;92-octane: 280 gal
E)​​92-octane: 280 gal;87-octane: 89 gal

Question 38

Solve the system of linear equations by the method of elimination.Find (x,y)and check your solution algebraically.​ $$\left\{ \begin{array} { l } 3 x + 2 y = 10 \\2 x + 5 y = 3\end{array} \right.$$ ​

A)(4,1)
B)(4,-4) ​
C)(-4,-1) ​
D)(4,-1) ​
E)(-4,1) ​

Question 39

Seven hundred gallons of 91-octane gasoline is obtained by mixing 87-octane gasoline with 94-octane gasoline.Write a system of equations in which one equation represents the amount of final mixture required and the other represents the amounts of 87- and 94-octane gasolines in the final mixture.Let x and y represent the numbers of gallons of 87-octane and 94-octane gasolines,respectively. ​

A)​ $$\left\{ \begin{array} { c } x + y = 85 \\87 x + 94 y = 63,700\end{array} \right.$$
B)​ $$\left\{ \begin{array} { c } 87 x + 94 y = 700 \\x + y = 85\end{array} \right.$$
C)​ $$\left\{ \begin{array} { l } x + y = 63,700 \\87 x + 94 y = 85\end{array} \right.$$
D)​ $$\left\{ \begin{array} { c } 87 x + 94 y = 700 \\x + y = 63,700\end{array} \right.$$
E)​ $$\left\{ \begin{array} { c } x + y = 700 \\87 x + 94 y = 63,700\end{array} \right.$$

Question 40

Solve the system of linear equations by the method of elimination.Find (u,v)and check your solution algebraically.​ $$\left\{ \begin{array} { l } 5 u + 6 v = 28 \\3 u + 5 v = 20\end{array} \right.$$ ​

A)​ $\left( \frac { 7 } { 20 } , \frac { 7 } { 16 } \right)$
B) $\left( - \frac { 20 } { 7 } , \frac { 16 } { 7 } \right)$ ​
C) $\left( \frac { 20 } { 7 } , - \frac { 16 } { 7 } \right)$ ​
D) $\left( - \frac { 20 } { 7 } , - \frac { 16 } { 7 } \right)$ ​
E) $\left( \frac { 20 } { 7 } , \frac { 16 } { 7 } \right)$ ​

Question 41

​Solve the system of linear equations by the method of elimination,find (x,y). ​​ $$\left\{ \begin{array} { l } - 0.06 x + 0.01 y = 0.03 \\- 0.09 x - 0.04 y = 0.09\end{array} \right.$$ ​

A)​ $\left( - \frac { 11 } { 7 } , - \frac { 11 } { 9 } \right)$
B) $\left( - \frac { 14 } { 11 } , - \frac { 9 } { 11 } \right)$
C)​ $\left( - \frac { 7 } { 11 } , - \frac { 27 } { 11 } \right)$
D)No solution
E)​ $\left( - \frac { 7 } { 11 } , - \frac { 9 } { 11 } \right)$

Question 42

An airplane flying into a headwind travels 1128.00 miles in 4 hours and 42 minutes.On the return flight,the distance is traveled in 4 hours.Find the airspeed of the plane and the speed of the wind,assuming that both remain constant. ​

A)plane speed = 238 mph;wind speed = 250 mph
B)plane speed = 276 mph;wind speed = 21.00 mph
C)plane speed = 261.00 mph;wind speed = 21.00 mph
D)plane speed = 238 mph;wind speed = 21.00 mph
E)plane speed = 276 mph;wind speed = 268 mph

Question 43

​Solve the system of linear equations by the method of elimination,find (x,y). ​​ $$\left\{ \begin{array} { c } \frac { 9 } { 4 } x + \frac { 1 } { 4 } y = \frac { - 7 } { 4 } \\9 x + y = - 7\end{array} \right.$$ ​

A)​(4,-43)
B)(a,-7 - 9a)​(infinitely many solutions)
C)​(2,-25)
D)​No solutions
E)​(8,-79)

Question 44

​Solve the system of linear equations using any method,find (x,y). ​​ $$\left\{ \begin{array} { r } 9 x - 2 y = 1 \\- 3 x + 2 y = 7\end{array} \right.$$ ​

A)​ $\left( \frac { 3 } { 4 } , \frac { 2 } { 11 } \right)$
B)​ $\left( \frac { 8 } { 3 } , \frac { 11 } { 2 } \right)$
C)​No solution
D)​ $\left( \frac { 4 } { 3 } , \frac { 11 } { 2 } \right)$
E)​ $\left( \frac { 1 } { 3 } , \frac { 5 } { 2 } \right)$

Question 45

Find the equilibrium point (x,p)of the demand and supply equations.(The equilibrium point is the price p and number of units x that satisfy both the demand and supply equations. ) Demand
Supply
P = 129.00 - 0.07x
P = 0.2x - 60
​

A)​(200,115)
B)(700,80)
C)No solution
D)​(514,42.8)
E)(300,108)​

Question 46

A wildlife biologist has spent the past 5 years studying the relationship between the length (inches)and the average weight (lb)of the striped bass.His results are summarized in the table below. $$\begin{array} { | l | l | } \hline \begin{array} { l } \text { Length } \\\text { (inches) }\end{array} & \begin{array} { l } \text { Weight } \\\text { (lb) }\end{array} \\\hline 14 & 3 \\18 & 6 \\22 & 11 \\26 & 14 \\\hline\end{array}$$
Find the least squares regression line y = ax + b for the data in the table,where x represents the length of the fish in inches and y represents the weight in pounds.Use this regression line to estimate the average weight of a striped bass that is 21 inches in length.

A)y = 0.85(21)- 12.5 = 5.35 lb
B)y = 0.95(21)- 10.5 = 9.45 lb
C)y = 0.95(21)- 8.5 = 11.45 lb
D)y = 0.75(21)- 9.5 = 6.25 lb
E)y = 0.65(21)- 10.5 = 3.15 lb

Question 47

Solve the system of linear equations by the method of elimination,find (x,y). ​​ $$\left\{ \begin{array} { l } \frac { x + 23 } { 8 } + \frac { y + 10 } { 7 } = 3 \\x - 9 y = 20\end{array} \right.$$ ​

A)No solution
B)​(20 + 9a,a)(infinitely many solutions)
C)(-3,-7)
D)(38,2)
E)(-7,-3)

Question 48

​Solve the system of linear equations using any method,find (x,y). ​​ $\left\{ \begin{array} { l } 3 x + 8 y = 2 \\y = x - 5\end{array} \right.$ ​

A)​ $\left( \frac { 42 } { 11 } , - \frac { 13 } { 11 } \right)$
B)​ $\left( \frac { 11 } { 42 } , \frac { 11 } { 26 } \right)$
C)​ $\left( \frac { 42 } { 11 } , \frac { 26 } { 11 } \right)$
D)No solution
E)​ $\left( \frac { 84 } { 11 } , \frac { 78 } { 11 } \right)$

Question 49

During one performance of a local arts council's presentation of Fiddler on the Roof,the box office sold 225 tickets and collected $1638.If adult tickets sold for $9 and children's tickets sold for $6,how many of each type of ticket were sold? ​

A)adult tickets sold = 96;children's tickets sold = 129
B)adult tickets sold = 115;children's tickets sold = 148
C)adult tickets sold = 129;children's tickets sold = 96
D)adult tickets sold = 115;children's tickets sold = 100
E)adult tickets sold = 148;children's tickets sold = 115