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College Algebra 12th

R. David Gustafson, Jeff Hughes

Chapter 6

Linear Systems

Educators


Problem 1

Fill in the blanks.
A set of several equations with several variables is called a _____ of equations.

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Problem 2

Fill in the blanks.
Any set of numbers that satisfies each equation of a system is called a ______ of the system.

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Problem 3

Fill in the blanks.
If a system of equations has a solution, the system is ______.

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Problem 4

Fill in the blanks.
If a system of equations has no solution, the system is ______.

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Problem 5

Fill in the blanks.
If a system of equations has only one solution, the equations of the system are ______.

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Problem 6

Fill in the blanks.
If a system of equations has infinitely many solutions, the equations of the system are ______.

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Problem 7

Fill in the blanks.
The system $\left\{\begin{array}{l}x+y=5 \\ x-y=1\end{array}\right.$ _____ (consistent, inconsistent).

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Problem 8

Fill in the blanks.
The system $\left\{\begin{array}{l}x+y=5 \\ x+y=1\end{array}\right.$ (consistent, inconsistent).

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Problem 9

Fill in the blanks.
The equations of the system $\left\{\begin{array}{l}x+y=5 \\ 2 x+2 y=10\end{array}\right.$ are ______ (dependent, independent).

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Problem 10

Fill in the blanks.
The equations of the system $\left\{\begin{array}{l}x+y=5 \\ x-y=1\end{array}\right.$ are ______ (dependent, independent).

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Problem 11

Fill in the blanks.
The pair (1,3) ______ (is, is not) a solution of the system $\left\{\begin{array}{l}x+2 y=7 \\ 2 x-y=-1\end{array}\right.$.

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Problem 12

Fill in the blanks.
The pair (1,3) ______ (is, is not) a solution of the system $\left\{\begin{array}{l}3 x+y=6 \\ x-3 y=-8\end{array}\right.$.

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Problem 13

Solve each system of equations by graphing.
$$\left\{\begin{array}{l}y=-3 x+5 \\x-2 y=-3\end{array}\right.$$

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Problem 14

Solve each system of equations by graphing.
$$\left\{\begin{array}{l}x-2 y=-3 \\3 x+y=-9\end{array}\right.$$

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Problem 15

Solve each system of equations by graphing.
$$\left\{\begin{array}{l}3 x+2 y=2 \\-2 x+3 y=16\end{array}\right.$$

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Problem 16

Solve each system of equations by graphing.
$$\left\{\begin{array}{l}x+y=0 \\5 x-2 y=14\end{array}\right.$$

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Problem 17

Solve each system of equations by graphing.
$$\left\{\begin{array}{l}y=-x+5 \\3 x+3 y=30\end{array}\right.$$

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Problem 18

Solve each system of equations by graphing.
$$\left\{\begin{array}{l}x-3 y=-3 \\2 x-6 y=12\end{array}\right.$$

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Problem 19

Solve each system of equations by graphing.
$$\left\{\begin{array}{l}y=-x+6 \\5 x+5 y=30\end{array}\right.$$

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Problem 20

Solve each system of equations by graphing.
$$\left\{\begin{array}{l}2 x-y=-3 \\8 x-4 y=-12\end{array}\right.$$

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Problem 21

Use a graphing calculator to approximate the solutions of each system. Give answers to the nearest tenth.
$$\left\{\begin{array}{l}y=-5.7 x+7.8 \\y=37.2-19.1 x\end{array}\right.$$

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Problem 22

Use a graphing calculator to approximate the solutions of each system. Give answers to the nearest tenth.
$$\left\{\begin{array}{l}y=3.4 x-1 \\y=-7.1 x+3.1\end{array}\right.$$

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Problem 23

Use a graphing calculator to approximate the solutions of each system. Give answers to the nearest tenth.
$$\left\{\begin{array}{l}y=3.4 x-1 \\y=-7.1 x+3.1\end{array}\right.$$

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Problem 24

Use a graphing calculator to approximate the solutions of each system. Give answers to the nearest tenth.
$$\left\{\begin{array}{l}29 x+17 y=7 \\-17 x+23 y=19\end{array}\right.$$

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Problem 25

Use a graphing calculator to approximate the solutions of each system. Give answers to the nearest tenth.
$$\left\{\begin{array}{l}y=x-1 \\y=2 x\end{array}\right.$$

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Problem 26

Solve each system by substitution, if possible.
$$\left\{\begin{array}{l}y=2 x-1 \\x+y=5\end{array}\right.$$

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Problem 27

Solve each system by substitution, if possible.
$$\left\{\begin{array}{l}2 x+3 y=0 \\y=3 x-11\end{array}\right.$$

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Problem 28

Solve each system by substitution, if possible.
$$\left\{\begin{array}{l}2 x+y=3 \\y=5 x-11\end{array}\right.$$

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Problem 29

Solve each system by substitution, if possible.
$$\left\{\begin{array}{l}4 x+3 y=3 \\2 x-6 y=-1\end{array}\right.$$

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Problem 30

Solve each system by substitution, if possible.
$$\left\{\begin{array}{l}4 x+5 y=4 \8 x-15 y=3\end{array}\right.$$

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Problem 31

Solve each system by substitution, if possible.
$$\left\{\begin{array}{l}x+3 y=1 \\2 x+6 y=3\end{array}\right.$$

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Problem 32

Solve each system by substitution, if possible.
$$\left\{\begin{array}{l}x-3 y=14 \\3(x-12)=9 y\end{array}\right.$$

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Problem 33

Solve each system by substitution, if possible.
$$\left\{\begin{array}{l}y=3 x-6 \\x=\frac{1}{3} y+2\end{array}\right.$$

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Problem 34

Solve each system by substitution, if possible.
$$\left\{\begin{array}{l}3 x-y=12 \\y=3 x-12\end{array}\right.$$

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Problem 35

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}5 x-3 y=12 \\2 x-3 y=3\end{array}\right.$$

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Problem 36

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}2 x+3 y=8 \\-5 x+y=-3\end{array}\right.$$

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Problem 37

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}x-7 y=-11 \\8 x+2 y=28\end{array}\right.$$

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Problem 38

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}3 x+9 y=9 \\-x+5 y=-3\end{array}\right.$$

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Problem 39

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}3(x-y)=y-9 \\5(x+y)=-15\end{array}\right.$$

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Problem 40

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}2(x+y)=y+1 \\3(x+1)=y-3\end{array}\right.$$

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Problem 41

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}2=\frac{1}{x+y} \\2=\frac{3}{x-y}\end{array}\right.$$

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Problem 42

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}\frac{1}{x+y}=12 \\\frac{3 x}{y}=-4\end{array}\right.$$

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Problem 43

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}y+2 x=5 \\0.5 y=2.5-x\end{array}\right.$$

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Problem 44

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}-0.3 x+0.1 y=-0.1 \\6 x-2 y=2\end{array}\right.$$

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Problem 45

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}x+2(x-y)=2 \\3(y-x)-y=5\end{array}\right.$$

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Problem 46

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}3 x=4(2-y) \\3(x-2)+4 y=0\end{array}\right.$$

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Problem 47

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}x+\frac{y}{3}=\frac{5}{3} \\\frac{x+y}{3}=3-x\end{array}\right.$$

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Problem 48

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}x+\frac{y}{3}=\frac{5}{3} \\\frac{x+y}{3}=3-x\end{array}\right.$$

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Problem 49

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}\frac{3}{2} x+\frac{1}{3} y=2 \\\frac{2}{3} x+\frac{1}{9} y=1\end{array}\right.$$

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Problem 50

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}\frac{x+y}{2}+\frac{x-y}{5}=2 \\x=\frac{y}{2}+1\end{array}\right.$$

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Problem 51

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}\frac{x-y}{5}+\frac{x+y}{2}=6 \\\frac{x-y}{2}-\frac{x+y}{4}=3\end{array}\right.$$

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Problem 52

Solve each system by the elimination method, if possible.
$$\left\{\begin{array}{l}\frac{x-2}{5}+\frac{y+3}{2}=5 \\\frac{x+3}{2}+\frac{y-2}{3}=6\end{array}\right.$$

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Problem 53

Solve each system by any method.
$$\left\{\begin{array}{l}x+y+z=3 \\2 x+y+z=4 \\3 x+y-z=5\end{array}\right.$$

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Problem 54

Solve each system by any method.
$$\left\{\begin{array}{l}x-y-z=0 \\x+y-z=0 \\x-y+z=2\end{array}\right.$$

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Problem 55

Solve each system by any method.
$$\left\{\begin{array}{l}x-y+z=0 \\x+y+2 z=-1 \\-x-y+z=0\end{array}\right.$$

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Problem 56

Solve each system by any method.
$$\left\{\begin{array}{l}2 x+y-z=7 \\x-y+z=2 \\x+y-3 z=2\end{array}\right.$$

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Problem 57

Solve each system by any method.
$$\left\{\begin{array}{l}2 x+y=4 \\x-z=2 \\y+z=1\end{array}\right.$$

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Problem 58

Solve each system by any method.
$$\left\{\begin{array}{l}3 x+y+z=0 \\2 x-y+z=0 \\2 x+y+z=0\end{array}\right.$$

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Problem 59

Solve each system by any method.
$$\left\{\begin{array}{l}x+y+z=6 \\2 x+y+3 z=17 \\x+y+2 z=11\end{array}\right.$$

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Problem 60

Solve each system by any method.
$$\left\{\begin{array}{l}x+y+z=3 \\2 x+y+z=6 \\x+2 y+3 z=2\end{array}\right.$$

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Problem 61

Solve each system by any method.
$$\left\{\begin{array}{l}x+y+z=3 \\x+z=2 \\2 x+2 y+2 z=3\end{array}\right.$$

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Problem 62

Solve each system by any method.
$$\left\{\begin{array}{l}x+y+z=3 \\x+z=2 \\2x+y+2 z=5\end{array}\right.$$

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Problem 63

Solve each system by any method.
$$\left\{\begin{array}{l}x+2 y-z=2 \\2 x-y=-1 \\3 x+y+z=1\end{array}\right.$$

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Problem 64

Solve each system by any method.
$$\left\{\begin{array}{l}x+y=2 \\y+z=2 \\3 x+3 y=2\end{array}\right.$$

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Problem 65

Solve each system by any method.
$$\left\{\begin{array}{l}3 x+4 y+2 z=4 \\6 x-2 y+z=4 \\3 x-8 y-6 z=-3\end{array}\right.$$

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Problem 66

Solve each system by any method.
$$\left\{\begin{array}{l}x+y=2 \\y+z=2 \\x-z=0\end{array}\right.$$

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Problem 67

Solve each system by any method.
$$\left\{\begin{array}{l}2 x-y-z=0 \\x-2 y-z=-1 \\x-y-2 z=-1\end{array}\right.$$

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Problem 68

Solve each system by any method.
$$\left\{\begin{array}{l}x+3 y-z=5 \\3 x-y+z=2 \\2 x+y=1\end{array}\right.$$

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Problem 69

Solve each system by any method.
$$\left\{\begin{array}{l}(x+y)+(y+z)+(z+x)=6 \\(x-y)+(y-z)+(z-x)=0 \\x+y+2 z=4\end{array}\right.$$

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Problem 70

Solve each system by any method.
$$\left\{\begin{array}{l}(x+y)+(y+z)=1 \\(x+z)+(x+z)=3 \\(x-y)-(x-z)=-1\end{array}\right.$$

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Problem 71

Use systems of equations to solve each problem.
If Jonathan purchases two hamburgers and four orders of french fries for $\$ 8$ and Hannah purchases three hamburgers and two orders of fries for $\$ 8,$ what is the price of each item?

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Problem 72

Use systems of equations to solve each problem.
Hunter purchases two tennis rackets and four cans of tennis balls for \$102.Jana purchases three tennis rackets and two cans of tennis balls for $\$ 141 .$ What is the price of each item?

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Problem 73

Use systems of equations to solve each problem.
A farmer raises corn and soybeans on 350 acres of land. Because of expected prices at harvest time, he thinks it would be wise to plant 100 more acres of corn than of soybeans. How many acres of each does he plant?

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Problem 74

Use systems of equations to solve each problem.
There is an initiation fee to join the Pine River Country Club, as well as monthly dues. The total cost after 7 months' membership will be $\$ 3025$, and after $1 \frac{1}{2}$ years, $\$ 3850$. Find both the initiation fee and the monthly dues.

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Problem 75

Use systems of equations to solve each problem.
A General Jackson riverboat can travel 30 kilometers downstream in 3 hours and can make the return trip in 5 hours. Find the speed of the boat in still water.

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Problem 76

Use systems of equations to solve each problem.
A rectangular picture frame has a perimeter of 1900 centimeters and a width that is 250 centimeters less than its length. Find the area of the picture.

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Problem 77

Use systems of equations to solve each problem.
A metallurgist wants to make 60 grams of an alloy that is to be $34 \%$ copper. She that is 250 centimeters less than its length. Find the area of the picture.

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Problem 78

Use systems of equations to solve each problem.
The two weights shown will be in balance if the product of one weight and its distance from the fulcrum is equal to the product of the other weight and its distance from the fulcrum. Two weights are in balance when one is 2 meters and the other 3 meters from the fulcrum. If the fulcrum remained in the same spot and the weights were interchanged, the closer weight would need to be increased by 5 pounds to maintain balance. Find the weights.

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Problem 79

Use systems of equations to solve each problem.
A 112 -pound force can lift the 448 -pound load shown. If the fulcrum is moved 1 additional foot away from the load, a 192 -pound force is required. Find the length of the lever.

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Problem 80

Use systems of equations to solve each problem.
For a test question, a mathematics teacher wants to find two constants $a$ and $b$ such that the test item "Simplify $a(x+2 y)-b(2 x-y)^{\prime \prime}$ will have an answer of $-3 x+9 y .$ What constants $a$ and $b$ should the teacher use?

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Problem 81

Use systems of equations to solve each problem.
Rollowheel, Inc., can manufacture a pair of in-line skates for $\$ 43.53 .$ Daily fixed costs of manufacturing in-line skates amount to $\$ 742.72 .$ A pair of in-line skates can be sold for 889.95. Find equations expressing the expenses $E$ and the revenue $R$ as functions of $x$, the number of pairs manufactured and sold. At what production level will expenses equal revenues?

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Problem 82

Use systems of equations to solve each problem.
For its sales staff, a company offers two salary options. One is $\$ 326$ per week plus a commission of $3 \frac{1}{2} \%$ of sales. The other is $\$ 200$ per week plus $4 \frac{1}{4} \%$ of sales. Find equations that express incomes $I_{1}$ and $I_{2}$ as functions of sales $x$, and find the weekly sales level that produces equal salaries.

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Problem 83

Use systems of three equations in three variables to solve each problem.
A college student earns $\$ 198.50$ per week working three part-time jobs. Half of his 30-hour work week is spent cooking hamburgers at a fast-food chain, earning $\$ 5.70$ per hour. In addition, the student earns $\$ 6.30$ per hour working at a gas station and $\$ 10$ per hour doing janitorial work. How many hours per week does the student work at each job?

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Problem 84

Use systems of three equations in three variables to solve each problem.
A woman invested a $\$ 22,000$ rollover IRA account in three banks paying $5 \%$ $6 \%,$ and $7 \%$ annual interest. She invested $\$ 2000$ more at $6 \%$ than at $5 \% .$ The total annual interest she earned was $\$ 1370 .$ How much did she invest at each rate?

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Problem 85

Use systems of three equations in three variables to solve each problem.
Approximately 3 million people live in costa Rica. 2.61 million are younger than 50 years, and 1.95 million are older than 14 years. How many people are in each of the categories 0 14 years, 15- 49 years, and 50 years and older?

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Problem 86

Use systems of three equations in three variables to solve each problem.
The engineer designing a parabolic arch knows that its equation has the form $y=a x^{2}+b x+c .$ Use the information in the illustration to find $a, b,$ and $c$ Assume that the distances are given in feet. (Hint: The coordinates of points on the parabola satisfy its equation.)

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Problem 87

Use systems of three equations in three variables to solve each problem.
The sum of the angles of a triangle is $180^{\circ} .$ In a certain triangle, the largest angle is $20^{\circ}$ greater than the sum of the other two and is $10^{\circ}$ greater than 3 times the smallest. How large is each angle?

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Problem 88

Use systems of three equations in three variables to solve each problem.
The path of a thrown object is a parabola with the equation $f(x)=a x^{2}+b x+c$ Use the information in the illustration to find $a, b,$ and $c .$ (Distances are in feet.)

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Problem 89

If no method is stated, describe how you would determine the most efficient method to use to solve a linear system.

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Problem 90

Describe how a system of three equations in three variables can be reduced to a system of two equations and two variables.

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Problem 91

When using the elimination method, how can you tell whether the system has no solution?

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Problem 92

When using the elimination method, how can you tell whether the system has infinitely many solutions?

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Problem 93

Use a graphing calculator to attempt to find the solution of the system $\left\{\begin{array}{l}x-8 y=-51 \\ 3 x-25 y=-160\end{array}\right.$.

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Problem 94

Solve the system of Exercise 93 algebraically. Which method is easier, and why?

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Problem 95

Use a graphing calculator to attempt to find the solution of the system $\left\{\begin{array}{l}17 x-23 y=-76 \\ 29 x+19 y=-278\end{array}\right.$

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Problem 96

Solve the system of Exercise 95 algebraically. Which method is easier, and why?

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Problem 97

Write a system of two equations in two variables with the solution (-2,5).

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Problem 98

Write a system of three equations in three variables with the solution $(-4,5,1)$.

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Problem 99

Write a system of two equations in two variables with no solution.

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Problem 100

Write a system of three equations in three variables with an infinite number of solutions.

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Problem 101

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
If a system of two equations in two variables is represented by two lines with the same slope and different $y$ -intercepts, then the system has an infinite number of solutions.

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Problem 102

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
If a system of two equations in two variables is represented by two lines with negative reciprocal slopes, then the system has an infinite number of solutions.

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Problem 103

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
If a linear system of three equations in three variables has infinitely many solutions, then any ordered triple is a solution of the system.

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Problem 104

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
When using the graphing method, a system of two equations in two variables can appear to have no solution and yet have a unique one.

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Problem 105

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
A linear system of two equations in three variables cannot have a unique solution.

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Problem 106

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
If a linear system of two equations in two variables has a solution set involving fractions, then use the graphing method to ensure accuracy.

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Problem 107

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
To solve a linear system of three equations in three variables, we use the graphing method.

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Problem 108

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
The system of equations $999 x-999 y=999$ and $-999 x+999 y=-999$ has an infinite number of =solutions.

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Problem 109

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
The system of equations $-777 x+777 y=-777$ and $777 x-777 y=-777$ has no solution.

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Problem 110

Determine if the statement is true or false. If the statement is false, then correct it and make it true.
The system of equations $555 x+555 y=555$ and $555 x-555 y=-555$ has no solution.

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