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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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Questions & Answers

what is the importance knowing the graph of circular functions?
Arabella Reply
can get some help basic precalculus
ismail Reply
What do you need help with?
Andrew
how to convert general to standard form with not perfect trinomial
Camalia Reply
can get some help inverse function
ismail
Rectangle coordinate
Asma Reply
how to find for x
Jhon Reply
it depends on the equation
Robert
whats a domain
mike Reply
The domain of a function is the set of all input on which the function is defined. For example all real numbers are the Domain of any Polynomial function.
Spiro
foci (–7,–17) and (–7,17), the absolute value of the differenceof the distances of any point from the foci is 24.
Churlene Reply
difference between calculus and pre calculus?
Asma Reply
give me an example of a problem so that I can practice answering
Jenefa Reply
x³+y³+z³=42
Robert
dont forget the cube in each variable ;)
Robert
of she solves that, well ... then she has a lot of computational force under her command ....
Walter
what is a function?
CJ Reply
I want to learn about the law of exponent
Quera Reply
explain this
Hinderson Reply
what is functions?
Angel Reply
A mathematical relation such that every input has only one out.
Spiro
yes..it is a relationo of orders pairs of sets one or more input that leads to a exactly one output.
Mubita
Is a rule that assigns to each element X in a set A exactly one element, called F(x), in a set B.
RichieRich
If the plane intersects the cone (either above or below) horizontally, what figure will be created?
Feemark Reply
can you not take the square root of a negative number
Sharon Reply
No because a negative times a negative is a positive. No matter what you do you can never multiply the same number by itself and end with a negative
lurverkitten
Actually you can. you get what's called an Imaginary number denoted by i which is represented on the complex plane. The reply above would be correct if we were still confined to the "real" number line.
Liam
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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