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      0.5 x 0.5 y = 10   0.2 y + 0.2 x = 4        0.1 x + 0.1 z = 2

( 10 , 10 , 10 )

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5 x + 3 y z = 5 3 x 2 y + 4 z = 13 4 x + 3 y + 5 z = 22

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

No solutions exist.

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     2 x 3 y + z = −1           x + y + z = −4     4 x + 2 y 3 z = 33

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3 x + 2 y z = −10 x y + 2 z = 7 x + 3 y + z = −2

( 1 , 2 , 3 )

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3 x + 4 z = −11 x 2 y = 5 4 y z = −10

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

( x , 8 x 5 , 14 x 5 )

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

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

Three odd numbers sum up to 61. The smaller is one-third the larger and the middle number is 16 less than the larger. What are the three numbers?

11, 17, 33

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A local theatre sells out for their show. They sell all 500 tickets for a total purse of $8,070.00. The tickets were priced at $15 for students, $12 for children, and $18 for adults. If the band sold three times as many adult tickets as children’s tickets, how many of each type was sold?

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Systems of Nonlinear Equations and Inequalities: Two Variables

For the following exercises, solve the system of nonlinear equations.

y = x 2 7 y = 5 x 13

( 2 , 3 ) , ( 3 , 2 )

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y = x 2 4 y = 5 x + 10

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x 2 + y 2 = 16 y = x 8

No solution

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x 2 + y 2 = 25 y = x 2 + 5

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x 2 + y 2 = 4 y x 2 = 3

No solution

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For the following exercises, graph the inequality.

For the following exercises, graph the system of inequalities.

x 2 + y 2 + 2 x < 3 y > x 2 3

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x 2 2 x + y 2 4 x < 4 y < x + 4

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x 2 + y 2 < 1 y 2 < x

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Partial Fractions

For the following exercises, decompose into partial fractions.

2 x + 6 x 2 + 3 x + 2

2 x + 2 , 4 x + 1

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10 x + 2 4 x 2 + 4 x + 1

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7 x + 20 x 2 + 10 x + 25

7 x + 5 , 15 ( x + 5 ) 2

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x 18 x 2 12 x + 36

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x 2 + 36 x + 70 x 3 125

3 x 5 , 4 x + 1 x 2 + 5 x + 25

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5 x 2 + 6 x 2 x 3 + 27

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x 3 4 x 2 + 3 x + 11 ( x 2 2 ) 2

x 4 ( x 2 2 ) , 5 x + 3 ( x 2 2 ) 2

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4 x 4 2 x 3 + 22 x 2 6 x + 48 x ( x 2 + 4 ) 2

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Matrices and Matrix Operations

For the following exercises, perform the requested operations on the given matrices.

A = [ 4 2 1 3 ] , B = [ 6 7 3 11 2 4 ] , C = [ 6 7 11 2 14 0 ] , D = [ 1 4 9 10 5 7 2 8 5 ] , E = [ 7 14 3 2 1 3 0 1 9 ]

4 A

[ 16 8 4 12 ]

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B + C

undefined; dimensions do not match

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B A

undefined; inner dimensions do not match

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C B

[ 113 28 10 44 81 41 84 98 42 ]

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E D

[ 127 74 176 2 11 40 28 77 38 ]

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C E

undefined; inner dimensions do not match

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Solving Systems with Gaussian Elimination

For the following exercises, write the system of linear equations from the augmented matrix. Indicate whether there will be a unique solution.

[ 1 0 −3 0 1 2 0 0 0    |    7 −5 0 ]

x 3 z = 7 y + 2 z = 5 with infinite solutions

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[ 1 0 5 0 1 −2 0 0 0    |    −9 4 3 ]

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For the following exercises, write the augmented matrix from the system of linear equations.

2 x + 2 y + z = 7 2 x 8 y + 5 z = 0 19 x 10 y + 22 z = 3

[ 2 2 1 2 8 5 19 10 22    |    7 0 3 ]

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4 x + 2 y 3 z = 14 12 x + 3 y + z = 100 9 x 6 y + 2 z = 31

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x + 3 z = 12 x + 4 y = 0 y + 2 z = 7

[ 1 0 3 −1 4 0 0 1 2    |    12 0 −7 ]

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

3 x 4 y = 7 6 x + 8 y = 14

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3 x 4 y = 1 6 x + 8 y = 6

No solutions exist.

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1.1 x 2.3 y = 6.2 5.2 x 4.1 y = 4.3

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2 x + 3 y + 2 z = 1 4 x 6 y 4 z = 2 10 x + 15 y + 10 z = 0

No solutions exist.

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x + 2 y 4 z = 8 3 y + 8 z = 4 7 x + y + 2 z = 1

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Solving Systems with Inverses

For the following exercises, find the inverse of the matrix.

[ 0.2 1.4 1.2 0.4 ]

1 8 [ 2 7 6 1 ]

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[ 1 2 1 2 1 4 3 4 ]

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[ 12 9 6 1 3 2 4 3 2 ]

No inverse exists.

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Practice Key Terms 2

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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