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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

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
find the 15th term of the geometric sequince whose first is 18 and last term of 387
Jerwin Reply
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
virgelyn Reply
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
kinnecy Reply
can someone help me with some logarithmic and exponential equations.
Jeffrey Reply
sure. what is your question?
ninjadapaul
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
ninjadapaul
I don't understand what the A with approx sign and the boxed x mean
ninjadapaul
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
ninjadapaul
oops. ignore that.
ninjadapaul
so you not have an equal sign anywhere in the original equation?
ninjadapaul
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
Idrissa Reply
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
Kevin Reply
a perfect square v²+2v+_
Dearan Reply
kkk nice
Abdirahman Reply
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
Kim Reply
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
Embra Reply
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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