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For the following exercises, find the solutions by computing the inverse of the matrix.

0.3 x 0.1 y = 10 0.1 x + 0.3 y = 14

( 20 , 40 )

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0.4 x 0.2 y = 0.6 0.1 x + 0.05 y = 0.3

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4 x + 3 y 3 z = 4.3 5 x 4 y z = 6.1 x + z = 0.7

( 1 , 0.2 , 0.3 )

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2 x 3 y + 2 z = 3 x + 2 y + 4 z = 5 2 y + 5 z = 3

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For the following exercises, write a system of equations to solve each problem. Solve the system of equations.

Students were asked to bring their favorite fruit to class. 90% of the fruits consisted of banana, apple, and oranges. If oranges were half as popular as bananas and apples were 5% more popular than bananas, what are the percentages of each individual fruit?

17% oranges, 34% bananas, 39% apples

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A sorority held a bake sale to raise money and sold brownies and chocolate chip cookies. They priced the brownies at $2 and the chocolate chip cookies at $1. They raised $250 and sold 175 items. How many brownies and how many cookies were sold?

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Solving Systems with Cramer's Rule

For the following exercises, find the determinant.

| 0.2 0.6 0.7 1.1 |

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| 1 4 3 0 2 3 0 0 3 |

6

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For the following exercises, use Cramer’s Rule to solve the linear systems of equations.

4 x 2 y = 23 5 x 10 y = 35

( 6 , 1 2 )

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0.2 x 0.1 y = 0 0.3 x + 0.3 y = 2.5

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0.5 x + 0.1 y = 0.3 0.25 x + 0.05 y = 0.15

( x , 5 x + 3)

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x + 6 y + 3 z = 4 2 x + y + 2 z = 3 3 x 2 y + z = 0

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4 x 3 y + 5 z = 5 2 7 x 9 y 3 z = 3 2 x 5 y 5 z = 5 2

( 0 , 0 , 1 2 )

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3 10 x 1 5 y 3 10 z = 1 50 1 10 x 1 10 y 1 2 z = 9 50 2 5 x 1 2 y 3 5 z = 1 5

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Practice test

Is the following ordered pair a solution to the system of equations?

5 x y = 12 x + 4 y = 9 with ( 3 , 3 )

Yes

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For the following exercises, solve the systems of linear and nonlinear equations using substitution or elimination. Indicate if no solution exists.

1 2 x 1 3 y = 4 3 2 x y = 0

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1 2 x 4 y = 4 2 x + 16 y = 2

No solutions exist.

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5 x y = 1 10 x + 2 y = 2

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4 x 6 y 2 z = 1 10 x 7 y + 5 z = 1 4 3 x + 6 y 9 z = 6 5

1 20 ( 10 , 5 , 4 )

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x + z = 20 x + y + z = 20 x + 2 y + z = 10

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5 x 4 y 3 z = 0 2 x + y + 2 z = 0 x 6 y 7 z = 0

( x , 16 x 5 13 x 5 )

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y = x 2 + 2 x 3 y = x 1

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y 2 + x 2 = 25 y 2 2 x 2 = 1

( 2 2 , 17 ) , ( 2 2 , 17 ) , ( 2 2 , 17 ) , ( 2 2 , 17 )

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For the following exercises, graph the following inequalities.

x 2 + y 2 > 4 y < x 2 + 1

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For the following exercises, write the partial fraction decomposition.

8 x 30 x 2 + 10 x + 25

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13 x + 2 ( 3 x + 1 ) 2

5 3 x + 1 2 x + 3 ( 3 x + 1 ) 2

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x 4 x 3 + 2 x 1 x ( x 2 + 1 ) 2

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For the following exercises, perform the given matrix operations.

5 [ 4 9 2 3 ] + 1 2 [ 6 12 4 8 ]

[ 17 51 8 11 ]

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[ 1 4 7 2 9 5 12 0 4 ]     [ 3 4 1 3 5 10 ]

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[ 1 2 1 3 1 4 1 5 ] 1

[ 12 20 15 30 ]

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det | 1 2 1 2 0 1 2 0 1 2 0 1 2 0 |

1 8

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If det ( A ) = −6 , what would be the determinant if you switched rows 1 and 3, multiplied the second row by 12, and took the inverse?

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Rewrite the system of linear equations as an augmented matrix.

14 x 2 y + 13 z = 140 2 x + 3 y 6 z = 1 x 5 y + 12 z = 11

[ 14 2 13 2 3 6 1 5 12    |    140 1 11 ]

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Rewrite the augmented matrix as a system of linear equations.

[ 1 0 3 2 4 9 6 1 2 | 12 5 8 ]
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For the following exercises, use Gaussian elimination to solve the systems of equations.

x 6 y = 4 2 x 12 y = 0

No solutions exist.

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2 x + y + z = 3 x 2 y + 3 z = 6 x y z = 6

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For the following exercises, use the inverse of a matrix to solve the systems of equations.

4 x 5 y = 50 x + 2 y = 80

( 100 , 90 )

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1 100 x 3 100 y + 1 20 z = 49 3 100 x 7 100 y 1 100 z = 13 9 100 x 9 100 y 9 100 z = 99

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For the following exercises, use Cramer’s Rule to solve the systems of equations.

200 x 300 y = 2 400 x + 715 y = 4

( 1 100 , 0 )

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0.1 x + 0.1 y 0.1 z = 1.2 0.1 x 0.2 y + 0.4 z = 1.2 0.5 x 0.3 y + 0.8 z = 5.9

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For the following exercises, solve using a system of linear equations.

A factory producing cell phones has the following cost and revenue functions: C ( x ) = x 2 + 75 x + 2 , 688 and R ( x ) = x 2 + 160 x . What is the range of cell phones they should produce each day so there is profit? Round to the nearest number that generates profit.

32 or more cell phones per day

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A small fair charges $1.50 for students, $1 for children, and $2 for adults. In one day, three times as many children as adults attended. A total of 800 tickets were sold for a total revenue of $1,050. How many of each type of ticket was sold?

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Questions & Answers

find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
An investment account was opened with an initial deposit of $9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
Kala Reply
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
Moses Reply
12, 17, 22.... 25th term
Alexandra Reply
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Shirleen Reply
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
hi vedant can u help me with some assignments
Solomon
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
Practice Key Terms 2

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Source:  OpenStax, College algebra. OpenStax CNX. Feb 06, 2015 Download for free at https://legacy.cnx.org/content/col11759/1.3
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