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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

how can are find the domain and range of a relations
austin Reply
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
how to understand calculus?
Jenica Reply
with doing calculus
SLIMANE
Thanks po.
Jenica
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
it is better to use unit circle than triangle .triangle is only used for acute angles but you can begin with. Download any application named"unit circle" you find in it all you need. unit circle is a circle centred at origine (0;0) with radius r= 1.
SLIMANE
What is domain
johnphilip
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
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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