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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

what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply
a man walks up 200 meters along a straight road whose inclination is 30 degree.How high above the starting level is he?
Lhorren Reply
100 meters
Kuldeep
Find that number sum and product of all the divisors of 360
jancy Reply
answer
Ajith
exponential series
Naveen
yeah
Morosi
prime number?
Morosi
what is subgroup
Purshotam Reply
Prove that: (2cos&+1)(2cos&-1)(2cos2&-1)=2cos4&+1
Macmillan Reply
Practice Key Terms 2

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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