Deck 41: Linear and Nonlinear Systems of Equations

Question 1

Solve the system by the method of substitution.Check your solution(s)graphically.​ $$\left\{ \begin{array} { r } 7 x + y = 3 \\x ^ { 3 } - 3 + y = 0\end{array} \right.$$ ​ ​

A) $( \sqrt { 7 } , 3 + 7 \sqrt { 7 } ) , ( 0,3 ) , ( \sqrt { 7 } , 3 - 7 \sqrt { 7 } )$
B) $( - \sqrt { 7 } , 3 + 7 \sqrt { 7 } ) , ( 0 , - 3 ) , ( \sqrt { 7 } , 3 - 7 \sqrt { 7 } )$
C) $( - \sqrt { 7 } , 3 - 7 \sqrt { 7 } ) , ( 0,3 ) , ( \sqrt { 7 } , 3 - 7 \sqrt { 7 } )$
D) $( - \sqrt { 7 } , 3 + 7 \sqrt { 7 } ) , ( 0,3 ) , ( \sqrt { 7 } , 3 + 7 \sqrt { 7 } )$
E) $( - \sqrt { 7 } , 3 + 7 \sqrt { 7 } ) , ( 0,3 ) , ( \sqrt { 7 } , 3 - 7 \sqrt { 7 } )$

Answer 1

$( - \sqrt { 7 } , 3 + 7 \sqrt { 7 } ) , ( 0,3 ) , ( \sqrt { 7 } , 3 - 7 \sqrt { 7 } )$

Question 2

Select the ordered pair that is a solution of the system of equations.​ $$\left\{ \begin{aligned}- \log x + 3 & = 2 \\\frac { 1 } { 9 } x + y & = \frac { 28 } { 9 }\end{aligned} \right.$$ ​

A) $\left( \frac { 1 } { 2 } , 10 \right)$
B)(10,2)
C)(10,28)
D) $\left( \frac { 1 } { 28 } , 10 \right)$
E)(2,10)

Answer 2

(10,2)

Question 3

Solve the system graphically.​ $$\left\{ \begin{aligned}x ^ { 2 } + y ^ { 2 } & = 85 \\( x - 5 ) ^ { 2 } + y ^ { 2 } & = 50\end{aligned} \right.$$ ​ ​

A)(8,-8), (8,8)
B)(-8,-8), (-8,8)
C)(-6,-7), (6,7)
D)(8,7), (8,8)
E)(6,7), (6,-7)

Answer 3

(6,7), (6,-7)

Question 4

Solve the system by the method of substitution.Check your solution(s)graphically.​ $$\left\{ \begin{array} { r } 2 x + y = 9 \\- x + y = 0\end{array} \right.$$ ​

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

Question 5

Solve the system graphically.​ $$\left\{ \begin{array} { l } x - y = 0 \\5 x - 2 y = 9\end{array} \right.$$

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

Question 6

Solve the system by the method of substitution.​ $$\left\{ \begin{aligned}x - y & = 4 \\7 x - 4 y & = 37\end{aligned} \right.$$ ​

A)(7,3)
B)(7,-3)
C)(-8,-4)
D)(-7,3)
E)(8,4)

Question 7

Solve the system by the method of substitution.​ $$\left\{ \begin{array} { l } 0.5 x + 3.2 y = 5.3 \\0.3 x - 1.4 y = - 1.8\end{array} \right.$$ ​

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

Question 8

Solve the system by the method of substitution.​ $$\left\{ \begin{array} { l } 1.3 x + 0.6 y = 5.7 \\0.9 x - 0.2 y = 2.1\end{array} \right.$$ ​

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

Question 9

Solve the system graphically.​ $$\left\{ \begin{aligned}x - y & = 1 \\6 x - 3 y & = 24\end{aligned} \right.$$ ​ ​

A)(-8,-7)
B)(-7,-6)
C)(7,6)
D)(7,7)
E)(8,7)

Question 10

Solve the system by the method of substitution.​ $$\left\{ \begin{array} { l } 2 x - y + 2 = 0 \\4 x + y - 5 = 0\end{array} \right.$$ ​ ​

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

Question 11

Select the ordered pair that is a solution of the system of equations.​ $$\left\{ \begin{array} { c } 2 x ^ { 2 } + y = - 11 \\- x - y = 12\end{array} \right.$$ ​

A)(1,13)
B)(13,-1)
C)(-1,-13)
D)(1,-13)
E)(13,1)

Question 12

Solve the system by the method of substitution.Check your solution(s)graphically.​ $$\left\{ \begin{array} { r } x ^ { 2 } + 2 y = 0 \\x ^ { 2 } - 6 x - y = 0\end{array} \right.$$ ​ ​

A)(0,4), (0,-8)
B)(0,0), (4,8)
C)(0,0), (-4,-8)
D)(4,0), (0,-8)
E)(0,0), (4,-8)

Question 13

Use a graphing utility to solve the system of equations.Find the solution(s)accurate to two decimal places.​ $$\left\{ \begin{aligned}y & = e ^ { x + 4 } \\x - y + 5 & = 0\end{aligned} \right.$$ ​

A)​ (-4,1)
B)​ (0,4)
C)​ (0,-4)
D)​ (4,4)
E)​ (-4,1)

Question 14

Select the ordered pair that is a solution of the system of equations.​ $$\left\{ \begin{array} { l } 2 x - y = 3 \\8 x + y = - 18\end{array} \right.$$

A)(-6, $- \frac { 3 } { 2 }$ )
B)(3,6)
C)( $- \frac { 3 } { 2 }$ ,-6)
D)( $\frac { 3 } { 2 }$ ,6)
E)(-3,-6)

Question 15

Solve the system graphically.​ $$\left\{ \begin{aligned}6 x + 8 y & = 48 \\x - 8 y & = - 20\end{aligned} \right.$$ ​ ​

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

Question 16

Solve the system graphically.​ $$\left\{ \begin{array} { r } - x + 2 y = 3 \\x - y = 2\end{array} \right.$$ ​ ​

A)(-7,-5)
B)(9,-9)
C)(7,5)
D)(9,5)
E)(-9,-9)

Question 17

Solve the system by the method of substitution.Check your solution(s)graphically.​ $$\left\{ \begin{aligned}x - y & = - 3 \\x ^ { 2 } - y & = 3\end{aligned} \right.$$ ​​ ​

A)(3,6), (1,-2)
B)(5,5), (1,-2)
C)(6,3), (-3,6)
D)(3,6), (-2,1)
E)(5,5), (-2,1)

Question 18

Solve the system by the method of substitution.Check your solution(s)graphically.​ $$\left\{ \begin{array} { l } x - 2 y = - 9 \\x + 3 y = 1\end{array} \right.$$ ​ ​

A)(-5,2)
B)(-3,-2)
C)(-5,-2)
D)(-3,-3)
E)(-3,2)

Question 19

Select the ordered pair that is a solution of the system of equations.​ $$\left\{ \begin{aligned}y & = - 6 e ^ { x } \\- 7 x - y & = 6\end{aligned} \right.$$ ​

A)(0,-6)
B)(0,6)
C) $\left( \frac { 1 } { 6 } , 0 \right)$
D) $\left( - \frac { 1 } { 6 } , 0 \right)$
E)(-6,0)

Question 20

Solve the system by the method of substitution.​ $$\left\{ \begin{aligned}6 x + 5 y & = - 6 \\- x - \frac { 5 } { 6 } y & = - 3\end{aligned} \right.$$ ​

A)No solution
B)(6,3)
C)(-5,-3)
D)(5,4)
E)(-5,-4)

Question 21

The weekly rentals for a newly released DVD of an animated film at a local video store decreased each week.At the same time,the weekly rentals for a newly released DVD of a horror film increased each week.Models that approximate the weekly rentals R for each DVD are​ $$\left\{ \begin{array} { l l } R = 380 - 24 x & \text { Animated film } \\R = 28 + 20 x & \text { Horror film }\end{array} \right.$$ ​ where x represents the number of weeks each DVD was in the store,with x = 1 corresponding to the first week.After how many weeks will the rentals for the two movies be equal?
​

A)10 weeks
B)6 weeks
C)4 weeks
D)12 weeks
E)8 weeks

Question 22

Solve the system graphically or algebraically.Find the solution(s)accurate to three decimal places.​ $$\left\{ \begin{array} { r } y - e ^ { - x } = 3 \\y - \ln x = 5\end{array} \right.$$ ​

A)(3.752,0.287)
B)(0.387,3.852)
C)(0.287,3.752)
D)(0.287,3.852)
E)(3.852,0.387)

Question 23

Find the dimensions of the rectangle meeting the specified condition. ​
The perimeter is 352 centimeters and the length is 4 centimeters greater than the width.
​

A)90 cm × 90 cm
B)90 cm × 86 cm
C)92 cm × 88 cm
D)94 cm × 90 cm
E)92 cm × 86 cm

Question 24

Solve the system graphically.​ $$\left\{ \begin{aligned}x - 3 y & = - 7 \\5 x + 3 y & = - 8\end{aligned} \right.$$ ​

A)(-3,-5)
B)(-5,3)
C) $\left( - \frac { 5 } { 2 } , - \frac { 3 } { 2 } \right)$
D) $\left( - \frac { 5 } { 2 } , \frac { 3 } { 2 } \right)$
E) $\left( - \frac { 5 } { 2 } , - 3 \right)$

Question 25

Solve the system graphically.​ $$\left\{ \begin{aligned}x + y & = 0 \\3 x - 2 y & = 10\end{aligned} \right.$$ ​

A)(2,0)
B)(2,-4)
C)(0,-2)
D)(2,-2)
E)(4,-2)

Question 26

What are the dimensions of a rectangular tract of land if its perimeter is 50 kilometers and its area is 150 square kilometers? ​

A)14 km × 19 km
B)10 km × 19 km
C)12 km × 15 km
D)10 km × 15 km
E)12 km × 17 km

Question 27

Solve the system by the method of substitution.​ $$\left\{ \begin{aligned}- \frac { 2 } { 3 } x + y & = 8 \\2 x - 3 y & = 20\end{aligned} \right.$$ ​

A)(-20,-40)
B)No solution
C)(20,40)
D) $\left( - \frac { 20 } { 3 } , \frac { 40 } { 3 } \right)$
E) $\left( \frac { 20 } { 3 } , \frac { 40 } { 3 } \right)$

Question 28

What are the dimensions of an isosceles right triangle with a four-inch hypotenuse and an area of 4 square inches? ​

A) $$2 \sqrt { 2 }$$ in.× $$2 \sqrt { 2 }$$ in.× 4 in.
B) $\sqrt { 10 }$ in.× $\sqrt { 10 }$ in.× 4 in.
C) $3$ in.× $3$ in.× 4 in.
D) $\sqrt { 11 }$ in.× $\sqrt { 11 }$ in.× 4 in.
E) $2 \sqrt { 3 }$ in.× $2 \sqrt { 3 }$ in.× 4 in.

Question 29

Find the dimensions of the rectangle meeting the specified condition. ​
The perimeter is 36 meters and the length is 4 meters greater than the width.
​

A)13 m × 7 m
B)13 m × 9 m
C)11 m × 7 m
D)15 m × 11 m
E)11 m × 11 m

Question 30

Find the dimensions of the rectangle meeting the specified condition. ​
The perimeter is 70 inches and the width is three-fourths the length.
​

A)22 in × 15 in
B)24 in × 19 in
C)22 in × 17 in
D)20 in × 19 in
E)20 in × 15 in

Question 31

Solve the system by the method of substitution. ​​ $$\left\{ \begin{array} { r } 9 x - 4 y - 2 = 0 \\x + 9 y - 4 = 0\end{array} \right.$$ ​

A)(0,0)
B)( $\frac { 2 } { 5 }$ , $\frac { 2 } { 5 }$ )
C)( $\frac { 2 } { 5 }$ ,- $\frac { 2 } { 5 }$ )
D)(- $\frac { 2 } { 5 }$ , $\frac { 2 } { 5 }$ )
E)(- $\frac { 2 } { 5 }$ ,- $\frac { 2 } { 5 }$ )

Question 32

Solve the system by the method of substitution.​ $$\left\{ \begin{aligned}x - y & = - 8 \\x ^ { 2 } - y & = - 21\end{aligned} \right.$$ ​

A) $\left( \frac { 25 } { 3 } , - \frac { 38 } { 3 } \right)$
B) $\left( - \frac { 25 } { 3 } , - \frac { 38 } { 3 } \right)$
C) $\left( - \frac { 25 } { 3 } , \frac { 38 } { 3 } \right)$
D)No solution
E) $\left( \frac { 25 } { 3 } , \frac { 38 } { 3 } \right)$

Question 33

A small software company invests $24,425 to produce a software package that will sell for $70.Each unit can be produced for $45.84.How many units must be sold to break even? ​

A)911 units
B)811 units
C)1211 units
D)1011 units
E)1111 units

Question 34

The total weekly sales for a newly released portable media player (PMP)increased each week.At the same time,the total weekly sales for another newly released PMP decreased each week.Models that approximate the total weekly sales S (in thousands of units)for each PMP are​ $$\left\{ \begin{array} { l l } S = 10 x + 70 & \text { PMP1 } \\S = - 10 x + 210 & \text { PMP2 }\end{array} \right.$$ ​ where x represents the number of weeks each PMP was in stores,with x = 0 corresponding to the PMP sales on the day each PMP was first released in stores.After how many weeks will the sales for the two PMPs be equal?
​

A)11 weeks
B)9 weeks
C)7 weeks
D)3 weeks
E)5 weeks

Question 35

Solve the system by the method of substitution.​ $\left\{ \begin{aligned}\frac { 1 } { 5 } x + \frac { 1 } { 2 } y & = 8 \\x + y & = 21\end{aligned} \right.$ ​

A) $\left( \frac { 25 } { 3 } , - \frac { 38 } { 3 } \right)$
B)(25,38)
C) $\left( - \frac { 25 } { 3 } , \frac { 38 } { 3 } \right)$
D) $\left( \frac { 25 } { 3 } , \frac { 38 } { 3 } \right)$
E)(-25,-38)

Question 36

Find the sales necessary to break even (R = C)for the cost C of producing x units and the revenue R obtained by selling x units.(Round to the nearest whole unit. )​ $$C = 6.5 \sqrt { x } + 9,674 , R = 3.2 x$$ ​

A)2,937 units
B)3,137 units
C)3,037 units
D)3,337 units
E)3,237 units

Question 37

Find the dimensions of the rectangle meeting the specified condition. ​
The perimeter is 528 feet and the length is $$4 \frac { 1 } { 2 }$$ times the width.
​

A)216 ft × 52 ft
B)216 ft × 48 ft
C)220 ft × 52 ft
D)218 ft × 48 ft
E)218 ft × 50 ft

Question 38

Find the sales necessary to break even (R = C)for the cost C of producing x units and the revenue R obtained by selling x units.(Round to the nearest whole unit. ) ​
C = 8630x + 264,000,R = 9950x
​

A)100 units
B)300 units
C)0 unit
D)400 units
E)200 units

Question 39

Find the dimensions of the rectangle meeting the specified condition. ​
The perimeter is 30.6 millimeters and the length is 2.4 times the width.
​

A)10.8 mm × 8.5 mm
B)12.8 mm × 4.5 mm
C)10.8 mm × 4.5 mm
D)14.8 mm × 8.5 mm
E)12.8 mm × 6.5 mm

Question 40

A small fast-food restaurant invests $5,951 to produce a new food item that will sell for $3.81.Each unit can be produced for $2.50.How many items must be sold to break even? ​

A)4543 items
B)4743 items
C)4343 items
D)4643 items
E)4443 items

Question 41

Solve the system by substitution,if possible.​ $$\left\{ \begin{array} { l } - 8 x + 7 y = - 11 \\16 x + 14 y = 106\end{array} \right.$$ ​

A)(​4,3)
B)​(16,14)
C)​​(- 4,- 3)
D)​(-11,106)
E)​(- 16,- 14)

Question 42

Find the dimensions of the rectangle meeting the specified conditions. The perimeter is 48 meters and the length is 2 meters more than the width.

A)13 m × 11 m
B)16 m × 14 m
C)16 m × 8 m
D)17 m × 15 m
E)12 m × 12 m

Question 43

Find the sales necessary to break even (R - C = 0)for the cost C of producing x units and the revenue R obtained by selling x units.(Round to the nearest whole unit. )​ $$C = 6.7 \sqrt { x } + 3000 , R = 8.1 x$$ ​

A)No real solution
B)387 units
C)355 units
D)377 units
E)355 units or 387 units

Question 44

Determine which ordered pair is a solution of the system.​ $$\left\{ \begin{array} { l } - 8 x + 7 y = - 10 \\- 9 x + 4 y = 12\end{array} \right.$$ ​

A)(3,-1)
B)(4,-6)
C)(-4,-6)
D)(-6,-4)
E)(-1,-3)

Question 45

Solve the system of equations by graphing.​ $$\left\{ \begin{array} { l } 4 x - 9 y = - 60 \\2 x + 9 y = 78\end{array} \right.$$ ​

A)(11,-4)
B)(-3,-8)
C)(3,8)
D)(-5,12)
E)(8,3)

Question 46

Solve the system by the method of substitution.​ $$\left\{ \begin{aligned}x - y & = - 22 \\x ^ { 2 } - y & = - 2\end{aligned} \right.$$ ​

A)(5,-13), (-4,-22)
B)No real solution
C)(5,-17), (-4,-26)
D)(5,27), (-4,18)
E)(5,-13)

Question 47

Solve the system by substitution,if possible.​ $$\left\{ \begin{array} { l } - 6 x + 7 y = 50 \\12 x + 14 y = 124\end{array} \right.$$ ​

A)(9,-6)
B)(1,8)
C)(-6,9)
D)(-7,10)
E)No solutions

Question 48

Solve the system by the method of substitution.​ $$\left\{ \begin{array} { l } y = - 4 x \\y = x ^ { 3 } + x ^ { 2 } - x\end{array} \right.$$ ​

A)(3,4)
B)(0,3)
C)(3,4), (-3,-4)
D)No real solution
E)(0,0)

Question 49

​Solve the system by the method of substitution. ​​ $$\left\{ \begin{array} { l } \frac { 1 } { 10 } x + \frac { 3 } { 10 } y = 1 \\\\- \frac { 1 } { 5 } x + \frac { 2 } { 5 } y = 1\end{array} \right.$$ ​

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

Question 50

Solve the system.​ $$\left\{ \begin{array} { l } 12 x - 28 y = 12 \\- 6 x + 14 y = - 6\end{array} \right.$$ ​

A)x = 3,y = 7
B)x = 4,y = 3
C) $$\left( x , \frac { 3 x } { 7 } - \frac { 3 } { 7 } \right)$$
D)x = 2,y = 3
E)No solution

Question 51

Determine which ordered pair is a solution of the system.​ $$\left\{ \begin{array} { l } - x + 9 y ^ { 2 } = 12 \\- 4 x + 9 y = 3\end{array} \right.$$ ​

A)(-3,1)
B)(-8,7)
C)(-1,3)
D)(-3,-1)
E)(-8,8)

Question 52

Solve the system of equations by graphing. ​​ $$\left\{ \begin{aligned}x + y & = 0 \\6 x - 7 y & = 13\end{aligned} \right.$$ ​

A)(-2,1)
B)(-1,1)
C)(0,2)
D)(1,-1)
E)(3,5)

Question 53

Use a graphing utility to solve the system of equations.Find the solution accurate to two decimal places.​ $$\left\{ \begin{array} { l } y = 2 e ^ { - x } \\y = \ln ( x - 2 ) + 1\end{array} \right.$$ ​

A)​(0.13,1.76)
B)​(- 1.40,0.49)and (- 0.25,1.56)
C)no real solution
D)​(4.77,0.07)
E)​(2.44,0.17)

Question 54

The sales of various types of lawn and garden tools vary according to the season.At a certain home improvement store,the monthly sales H of garden hoes (hoes sold per month)decline from July to October whereas the monthly sales of lawn rakes R (rakes sold per month)increase during this same interval.The sales of these two items during the calendar months July-October are modeled by the equations:​ $$\begin{array} { l } H ( t ) = 64 - 6 t \\R ( t ) = 17 t - 143\end{array}$$ ​ where t is the month (t = 7 corresponds to July).In which month does the number of rakes sold equal the number of hoes sold?
​

A)September
B)July
C)November
D)October
E)August

Question 55

Solve the system by substitution,if possible.​ $$\left\{ \begin{array} { l } y = 7 x - 21 \\9 x - y = 29\end{array} \right.$$ ​

A)(11,-2)
B)(7,4)
C)(-3,12)
D)(4,7)
E)No solutions

Question 56

Solve the system by substitution,if possible.​ $$\left\{ \begin{array} { l } 4 x + 3 y = 23 \\8 x + 6 y = 24\end{array} \right.$$ ​

A)(6,5)
B)(4,7)
C)(-5,-1)
D)(2,5)
E)No solution;inconsistent system

Question 57

​Solve the system of equations by graphing.​ $$\left\{ \begin{array} { l } 2 x + 3 y = 28 \\6 x + 5 y = 60\end{array} \right.$$ ​ ​

A)(-6,5)
B)​(3,- 5)
C)​(5,6)
D)​(2,6)
E)​(5,3)

Question 58

Solve the system by the method of substitution.​ $$\left\{ \begin{array} { l } y = x ^ { 3 } - 6 x ^ { 2 } + 4 \\y = x ^ { 2 } - 10 x + 4\end{array} \right.$$ ​

A)(2,-12), (1,-5)
B)(5,-21), (2,-12), (0,4)
C)(5,-21), (-1,15)
D)No real solution
E)(5,-21), (-2,28), (1,-5)